I hate to say it, but I can't really come up with a way to justify gryphons, or at least flying ones. Their body structure is such that flight seems impossible, and if they could fly they would have such hollow bones that they would be little threat on the ground (they lack strength and their bones would snap like twigs), making the idea of sharp claws and a hunter build rather impractical. I don't believe any plausible gryphon would be capable of flying.
Thus to justify them I look at land based animals that still use wings.
...This changed from lots of options to a slow progression to a final gryphon. You can skip to the bottom if you just want to see the end result.
Look at Ostrich as a land based winged bird for a starting format. They use wings for balance when running, not for flight. A similar approach could be taken for gryphons. They are land based predators (thus the teeth and claws) that evolved from birds but but now use wings entirely for maneuvering while running.
However, justifying four legs is hard here. Four legs provides balance and stability. Four legs and a tail does the same thing for balance as ostrich wings do. Wings on ostrich work by allowing an ostrich to stick to two legs while still having a means to control their balance and adjust their direction better than two legs alone allow; but ultimately they are an adaption needed only because two legs are less balanced then those with four.
I toyed with the idea of saying that they developed four legs but still prefer to run on two for speed, using only four legs when not at full speed, but this doesn't really work either. Two legs are more efficient for walking then four, in fact the efficiency of two legs is thought to be why humans adopted it.
One option is to mix in a little t-rex into your ostrich template. Maybe their front claws are not used for walking at all, but are grabbers used for catching prey. They run on two legs and grab their prey with front claws. You could perhaps have them then evolve to look somewhat similar to gryphons this way, but they would behave different. Ultimately it feels like a creature that is grpyhon in name only, even if at a distance they may pass for one.
lets move from t-rex to raptor! It's theorized that birds evolved from creatures sort of like raptors, and in fact raptors likely had feathers. In this case the raptors were believed to use their feathers and wing-like appendages to help them leap at prey and then cling on, balancing by flapping their proto-wings.
Something similar could be done with gryphons...sort of. Again the four legs become a hindrance to justify here. Four legged creatures would leap in a different way and would control their leap by twisting their body like cats. Wings would allow slightly more control, but not enough to really justify evolving them.
Tree Cat approach
One option I can think of is to have leaps not come from the ground, but from above. Imagine a scaled down gryphon that is something like a treecat. It lies in wait above it's prey and uses wings when it leaps down on it's prey to help guide it's fall and help it stay on it's prey while using all four limbs. This would require a very mobile prey that is difficult to stay on. Likely prey that is relatively large compared to the size of the gryphons, such that they can have four claws dug in and be using wings to help hold on to it.
This is a concept that could...somewhat work (honestly, the wings still don't quite seem useful enough to be justified), but explaining it's evolution is hard. If it started out as a bird the wings make more sense, they may be vestigial and while they were adapted to help with hunting they will eventually be adapted away from entirely to be more pure-cat. Their reason for evolving away the ability to fly could be because they tree cover was so extreme that it's hard to hunt land animals through the tree, and the extra mass and stronger bones they gain by giving up on flight allows them to have more lethal poncing power (and more ability to SURVIVE pouncing on something nimble that will try to buck them off and/or kill them) However, justifying the evolution of front paws is quite hard then. It would likely be that front paws started as some way to help them better cling to prey but it feels like a chicken and the egg problem. Once the front claws have reached a certain level of usefulness they're clearly advantageous, but they must be useful through all stages of evolution, not just the final one, and I don't see how the original proto-claws would ever be useful.
If you started with tree-cat format and tried to add wings later you have the problem that wings aren't all that useful compared to a well developed tree-cat. Watch a slow motion video of a cat leaping any large distance, they are really impressive at it without wings helping them. Again this is because the approach used to handle balance and leaping in a four limbed creature is different then the two limbed creature, as described with the ostrich example; rendering the wings less useful when four legs exist and hard to justify evolving towards.
There is also the problem that four limb creatures are larger and thus heavier. The heavier a creature the less wings help because it becomes harder to produce sufficient thrust from the wings to counteract the weight of their body enough to modify their flight/fall pattern.
One option is to try to focus on wings being needed primarily to help them stay on to a creature that they had pounced on, perhaps one that had evolved a thick or scaled coating that is resistant to their claws digging into. Or alternatively if they actually have to crawl along the length of a creature they already pounced on (by slowly moving a limb at a time) the presence of wings would help them stay on while they had fewer than four limbs dug into their prey.
Then we get the problem of how do they kill any prey that is large enough to require clinging on like this; where the cat's body is a small enough weight compared to the prey to be easily carried or bucked off. Maybe it climbs it's way to the neck of the prey and attacks with smaller cuts that are not instantly fatal but will kill the creature eventually. Imagine something only slightly larger than a cat pouncing on a deer, climbing along the deer to reach it's neck, digging in claws and flapping wings to hold on to the deer's frantic attempts to buck the cat off, eventually using it's claws (likely one specially adapted super-claw) to cut a long gash along the neck, or a few, and then clinging on while the prey slowly bleed out and died.
In fact, this could solve another problem, the beak. Beaks don't make much sense on land animals, they need their teeth to help rip meat off of a large prey (where as birds tend to eat their prey whole in one gulp). Perhaps the prey that the gryphon adapted to is hard to kill because of it's size, and a beak is developed to be able to kill the larger prey more effectively. It would be rather different then traditional falcon beaks, but look similar enough. Give it a hard pointy end with serrated edges and it could act like an extra long claw to cut at a throat while your cat holds on with all it's claws and wings. It would then have to be able to cut up the prey into beak-sized bites since it lacks teeth, but surely its claws or serrated beak could be used to cut off pieces.
This in turn leads to two questions, one, why is such a large prey animal in a forest so dense to make flying birds give up on flying (this is the most forgivable, and the prey doesn't have to be quite as big as a deer anyways), and two, why does our gryphon need to kill something so big? It can only eat so much food at once, killing such large prey is overkill, and hunting things larger than you increases the risk of being killed by the thing you're hunting.
One option is that the gryphons are pack hunters, with a large pack attacking the same prey animal, dividing the meat of the large prey across multiple animals so less meat is wasted. However, I'm not sure why they would benefit from being in a pack if they relied on ambush tactics like pouncing from above. There isn't much reason three attacking the same prey are more likely to secure a kill then three attacking their own separate prey. Perhaps if they kill not through an exact cut to the neck but through lots of smaller attacks that slowly weaken the creature there would be an advantage on having three or four all attacking, but frankly this seems unlikely since a cut to random non-vital areas are not likely to lead to quick kills, and because their wings would get in the way of each other if they were all fighting on the same creature.
Maybe their pack hunting consisted of spreading out to spot potential (semi-rare) prey and communicating it's presence over large distances to help set up an ambush by letting the attacking gryphon know where to attack. They benefit from the pack not because it helps take down the chosen prey, but by help finding and setting up the ambush for their prey. This helps a good bit, but it still doesn't seem a sufficient justification for living in a pack.
Perhaps if you also added in predators that prey on the gryphons as well, such that the pack behavior is as much about defending against those that would eat them as it is getting something to eat yourself the pack can be justified. Though the argument about them hunting by spreading out to find prey runs counter to the argument about them being in a pack for mutual protection. Maybe the pack is more about defending young that are particularly vulnerable and need a pack of non-hunters to defend them, while the fully grown hunters are mostly safe when working semi-independently to identify and work to help set up ambush of prey. This could work, packs form up for defense, then the males collaborate to kill larger prey to feed the many hungry beaks of the pack with a single very large kill. Cats are already well adapted to living on small number of successful large kills compared to other hunter species, which works well if much of hunting is about finding hard to find vulnerable prey; leaving one to the whims of chance as to when a kill can be obtained.
There is still the problem of chicken meets egg. Even if you assumed this approach worked once a fully formed gryphon existed (and honestly I would need to do some more work to justify the mini-gryphon final form, I think the basis could work but there are still a lot of holes to patch), you still have to ask how they reached that point. The creature isn't hunting large animals like this from start, it must have originally preyed on smaller species. More importantly, How did proto-wings prove advantages enough to stick around and develop into full wings? In my example above it's the flapping of wings that prove useful, until you develop a certain degree of strength in your wings they aren't too useful for staying on prey, so what good were they as proto-wings that justified their sticking around long enough to develop into full wings? well..maybe I can handle that too.
Flying Squirrel approach
Flying squirrel have their own wing-like appendages which they can spread to allow them to glide between trees. If we stuck with mini-gryphons as cats adapted to living within trees then perhaps you could claim that wings were used in much the same way that flying squirrels did, to help with gliding.
In this case imagine cats that not only hunt from trees, but live mostly within them. Presumably there are land predators much larger than them that are dangerous enough that gryphons avoid land whenever possible and travel from tree to tree. They have developed cats tremendous leaping ability and instincts, but even then traveling long distances between trees is difficult, particularly because if you miss your landing you go splat.
In this situation anything that would help the gryphon to focus his landing is going to allow him to leap further distances safely, which may be useful if tree cover is sometimes sparse. Even a small proto-wing produced by some bizarre adaptation accident may provide some minute additional potential to help land jumps.
This wouldn't be exactly flying squirrel approach, a predator is going to be too heavy to glide easily. Small protowings won't provide much lift to counteract weight. However, they may help with controlling and aim one's fall by acting more like small steering rudders to shift the direction and angle one is falling. Using conservation of momentum means opening and shutting anything during a twist could allow more control of how much the body rotates to align for a landing.
One of the problems I mentioned above was that gryphons couldn't hunt large land based prey as easily, so another option is to say that gryphons hunt other tree dwellers. Perhaps they used their wings when leaping at tree dwellers because those tree creatures could leap off after the gryphon leap and angle away from the gryphon. The presence of wings could help the gryphon to make better mid-air adjustment to their movement, and possible even slightly (emphasis on slightly) slow their descent if they and their prey end up falling to the ground after the pounce. This could be an intermediary step to the larger gryphon I mentioned above, or the final type of a smaller gryphon. The earlier version of gryphon I mentioned has the advantage of being larger and more predatory, likely with stronger wings developed to have strong flapping power instead of being predominately gliders, unlike the smaller 'flying squirrel' variant that preys on other tree creatures. The large raptor-cat thus is closer to a traditional gryphon's form. However, the larger gryphon is also a little harder to justify. You could even have both versions of gryphon in the same story.
Still, what was the original wing, before it was even much of a proto-wing you may ask. I'm thinking a mane....no really! Or at the minimum something that served the same use as a mane. Manes are actually detrimental to male lions in almost every way, costing energy to create and maintain, getting in the way during fights, and helping to trap heat against the body in sweltering savanna heat, and yet lions have them. Why? Because the ladies like it. Many species have developed costly traits that are apparently detrimental to males just to prove that the males can still survive despite the handicap, to prove just how fit they are, and thus how fit their daughters will be.
I'm thinking wings originally started as a 'mane' for cats, something males possess only to prove their fitness to females, despite the apparent cost, and which likely looked something like a mane. Likely the mane, like a peacock again, was also brighter colored than the rest of the body, their way of saying, look even with these bright colors that attract predators I can still survive, I'm just that good, and so will be my offspring.
However, as they moved into the trees these mane got even more 'in the way', catching on tree branches in mid jump can quickly be lethal, so the boys developed a trade off, show off their mane only when the women are looking and tuck them out of the way otherwise. Like our example peacock tails the proto-gryphons developed a way of hiding their 'mane' away when not in use, and flashing it only when they were showing off (which incidentally implies proto-gryphons would use a mating display approach similar to many birds? This in turn implies either the mane existed before pack life or packs consist of multiple males with females having choice of mates rather than a few males ruling a harem of females).
Only later did the males realize that by opening and closing their mane while leaping they could use it's size (by now it would be noticeably larger due to the Lensman Arms Race with other males to see who's was biggest (nothing Freudian here honest!). As time goes on it becomes common place for males to use their mating-display mane as a conservation of of angular momentum assistant.
From there the mane starts to adapt to move back to a more covenant location, closer to center of mass, to support males, and they start to develop the ability to open and close their mane/proto wing faster to support tricker leaps. About this time the mating display would have adapted away from just showing off their pretty mane and it's colors, instead focusing on showing off how they can flap the mane in a fast and interesting ways (again like many birds do with their wings).
Eventually the mane would start to show on females as well (originally as a sexually selective trait it would be predominately a male only trait, much as female lions lack manes), though the females would likely have cheap bland manes without caloric intensive colors. As time goes on the proto-wings usefulness in helping control leaps would become the primary use of the wing, with using it for courtship displays being less relevant (ironically because the protowings are now useful they become less valuable as a way of proving males fitness, for that the male needs to have entirely pointless limiting adaptions that he can still survive despite. Still, as long as the proto-wing was brighter colored than females it could serve this role).
Pulling it all together in just a few dozen millennia
This approach produces a smaller gryphon then traditional, but it's the only approach that seems viable to me (and it still needs a bit of hand-waves, the real answer is gryphons won't exist). If you wanted larger gryphons you could claim that my tree-gryphons evolved first and later some gryphons evolved to larger sizes and that their wings are now fully vestigial and simply haven't evolved away yet I suppose.
You start with a smaller cat, bigger than your traditional house cat but not significantly so, which lives in a forested area. It lives on land but is comfortable in trees as well, likely climbing them to avoid predators and sometimes to sneak up on small prey (which it could take on the ground, but surprising them avoided energy chasing them)
while still land base they developed a 'mane' which was a large tuft of hair on the back of their neck or upper part of their lower back that only males used, which was used for courtship displays and competitions much the same way that peacocks do.
Around this time some larger predators migrate in and start to make cat their favorite meal. At least one of these predators is likely sneaky and able to pounce on the cat before it can climb a tree to get away. To combat this the cats start spending more and more time in trees, sleeping and mating in trees but still hunting on the ground. Around this time the male develop a way to hide their mane by closing it down along the body to keep it out of the way of the tree branches they're now regularly traveling through. They can still open up the mane during display. The make likely starts to take on brighter colors now to maintain it's role as sexual indicator due to the "handicap principle"
Over time the cats start to develop more skill for traveling from branch to branch to allow travel while avoiding the increasingly dangerous ground. They still hunt on the ground, but they have become more adapt at finding prey from above and only going to the ground long enough to kill it, or pouncing on it from tree top. They may also hunt some smaller tree dwelling species, but the have trouble catching them because they aren't as nimble as native twee dwellers yet.
Some time later males realize that by opening up their manes while twisting during a mid air leap they can have affect their angular momentum, potentially helping them to land tricking jumps to far away branches without having to touch land, or even tricky pounces on a vulnerable tree-dweller. This proves advantageous and so quite quickly (from an evolutionary perspective) the manes grow longer and thicker, to have a larger effect on angular momentum, and the males better at 'releasing' quickly and controlling how much it expands to help with jumps. Courtship displays adjust to demonstrate the dexterity of manes. Around this time females start to develop manes as well without the bright colors.
The tree-cats (as they now effectively are) start preying more and more on fellow small tree creatures using their better pouncing ability, and enhanced ability to control their spin to adjust for tree creatures suddenly taking off after the cat leapt. They still prey on ground creatures as well, but since being on the ground is dangerous they carrying their prey into the tree as quickly as possible after a kill. They tend to focus on slightly larger land prey now since taking on land prey means a certain danger from land predators, so it's best to minimize the number of times you have to go to ground for food by bringing back the most caloric prey as you can as spoils of war.
The mane is now a proto-wing, used primarily for controlling angular momentum, not courtship. It has some flapping ability but not as much downward force. It's moving away from the base of the neck to the center of back and then splitting to cover the sides because these are the areas where angular momentum can best be controlled. Some 'flapping' ability is starting to develop.
A new predator enters the mix, or perhaps the old predator realizes all his prey are in the trees and an smaller offshoot species that can hunt in the trees evolve. This tree predator specifically target tree-cat young, since the treecat take time to develop to the point that they can safely move around trees. One mother has a heard time protecting against small but fast and/or sneaky predators that can attack from any direction (including above or below) of the trees; the predators like to slip in and steal a kitten the moment the mother is looking another direction, or worse hunting. To combat this the treecat females start living closer together, making it harder for predators since if any one female detects them it sets off an alarm and all the females start to hunt the would-be predators for dinner (remember, if they prey on kittens the predators are likely smaller than full grown adult tree-cats).
The mothers eventually develop a herd mentality, multiple mothers group together to create a parameter area where they keep their young. This changes the males and courtship. Now that females are already grouped together, and don't want to move around as much due to concern for leaving their developing young unprotected, courtship displays don't make as much sense. Instead your looking at migrating to a harem approach, with multiple males fighting for mating rights for one of these female herds, much like lions. This means that the mane-turned-wing no longer is useful for courtship, and it can now fully develop into a wing since that is it's sole use.
The herd starts to run into trouble with food. With so many proto-gryphons (yep, they now evolved past the tree-cat species) there aren't enough of the smaller prey species they use to hunt in the area of the heard (the more solatairy tree-cats held territories where they didn't' have to share their prey with other tree cats). Furthermore since the mothers need to stay close to their young to protect them they are less comfortable ranging as far to hunt.
Specialized hunting parties are created, so that some females can guard the kittens while others hunt and bring back food for kittens and guarding females. This could potentially lead to males becoming hunters and earning mating rights to females based off of their effectiveness in bringing in prey; or it could stick to the lion approach of only a few males in a pride and the females still hunting (it depends on your goals, if you want to shoot for sapience you want the males as hunters, more males in a social group and increased need to compete for mates on a social level will drive towards sapience). This division of labor will further the herding instincts, and particularly if you keep more then a single breeding male in the pride your quickly develop complex social cues and behaviors.
Eventually the need for more food drives the proto-gryphons to hunting larger and larger game because the small game doesn't provide enough resources to feed an entire pride of females and young that aren't hunting. In fact lets push the proto-gryphons to larger prey even harder. Maybe the same predators that threaten their kittens have done a good job of thinning out the smaller tree creatures, forcing the proto-gryphons to return to land based prey as the only numerous prey species left. We could even lower the number of smaller prey species on land (that are easily ambushed from trees) due to similar predators hunting them if we want to further drive the gryphons down my target path of hunting larger land animals. This could work quite well if you assume one new dangerous invasive species introduced to the old cat's world, the species that drove them to the trees and which, after splitting off into another tree based species later, continued to threaten the new tree-cats. New invasive species are a great driver of evolution, and help to explain why there are fewer small prey (those that didn't adapt to the new predators died out).
Gryphons are by this time built for precision leaping, not speed, so they focus on ambush from above tactics instead of the more traditional chasing and harassment techniques of most pack hunters. They develop more lethal claws and beak to help them kill prey that is noticeably larger than them.
At the same time the increased focus on social behaviors of pride would lead to social hunting techniques. Since ambushes don't work well for packs they focus on spreading out to find appropriate prey, large enough to provide lots of food, small or weak enough to kill. The inerrant dangers of predators on the ground drive gryphons to prioritizing the perfect selection of prey, they don't attack unless they're relatively assured of a kill and of sufficient calories being brought out of the kill to justify the risk. This is where the pride is so useful, they can scan much larger areas to find the perfect prey that is worth the risk. Once prey is found they use calls to alert others and communicate where prey is and where it should be best ambushed at. The finders of the prey they drive the prey towards an ambush spot, they do this by visibly positioning themselves to attack; if the prey doesn't move they driving gryphons will attack and kill it, if it does move odds are it's towards a preselected ambush spot. All of this is to allow setting up the most safest kill, ensuring a kill while decreasing odds of death to land predators or the (much larger) prey animal.
For the kill the gryphons will leap from above to get as close to a vulnerable neck of the gryphon as possible, using their impossible good leaping skills, the cats nimbleness in twisting their body midleap, and their wings (for balance and angular momentum primarily, not flight) to help them to make a uniquely precise leap. Once they hit their prey they dig in with all four paws to hold on.
Since a gryphon will never be able to perfectly aim a leap right for the neck they often need to finish the approach by 'climbing' to the neck may be needed. All the while they prey is trying to dislodge the gryphon as best it can, this is where the wings prove invaluable to keep from being thrown. Still, your need to get to the neck quickly or you will get dislodged or struck against trees; thus the emphasis on positioning the perfect ambush leap to get as close to a vulnerable spot as possible.
Once you get to the neck (or any other appropriate weak spot, the beak is used to land the killing blow, It's extended length gives the most flexibility in killing, you can keep all four claws safely digging into skin and the beak offers an extended reach; basically a long pointy and serrated beak gives a much larger radius from which a gryphon can kill so that if he lands anywhere close to the neck he can score a quick blow without the much harder task of trying to 'climb' toward his prey.
The human approach
An interesting side effect of this approach is that it's actually rather conducive to evolution of sapience! They live in a heard and hunt as a pack. However, they need rather extensive communication skills and long term planning to position the perfect ambush, both good drivers of intellect. They are taking on creatures larger and stronger, but can't evolve to be larger themselves since they need to 'fit' in the trees and need to stay small enough to exploit their wings; so the obvious solution is to focus on developing intelligence to take on the larger predators more intelligently. I imagine developing the use of projectiles to attack on land animals pretty quickly
Meanwhile, the social structure, having males mixed with females all trying to compete for mating rights, would be a large driver for intellect. The males will likely find making ambushes to be a great way to earn mating rights, but also highly dangerous; thus key planning is needed for deciding when it's worth the danger of an ambush to impress the females, and social skills are needed to get the pack to go along with your decision.
Finally, as a tree based creatures the gryphons would likely develop paws with the ability to grip the branches. This is the first step towards developing rudimentary thumbs, and thus the ability to work tools.