So, we have genetically engineered big cats with genetically inflated brains. However, these are cat brains, and not ape brains. Now, Ligers are crosses between lions and tigers, so while tigers are pretty much solitary, lions, or at least lionesses are quite social (the males are sociable to a lesser degree), so we can expect at least some of that sociability. I am presuming that Ligers' brains are simply scaled up big cat brains without any geneticist-imposed imperatives.
Cats do communicate with each other, despite being largely solitary, except for lions, and, we can presume, ligers. They would share the basic language of the cats from which they descended, of sounds and body language. However, with inflated brains would come inflated language centres, and additional complexity in communication is a certainty. That their communication would be quite unlike ape communication is also pretty much a certainty. While apes vocal communication takes the form of a variety of hoots and screeches, and this is modified to more modulated vocal communication, cats communicate vocally with more limited growls and roars. While this could be modulated to a human-intelligible voice, there is no reason for it to be so convenient for us.
Some bird calls sound like a fairly bland raucous squawk, but examine them using modern sound processing equipment, and you'll find that there are subtle, rapid frequency shifts. So might our Ligers have particularly resonant roars that are carriers for subtle frequency shifts and harmonics in which information can be conveyed. As cats are known for their particularly good hearing, they would have no problem receiving such communication.
So, we have communication that to a human ear still sounds like a roar, perhaps modulated a bit, but if a human tried to emulate that, it would sound to the Ligers like a monkey trying to emulate a human voice would sound to humans - amusing or infantile at best.
As for the particulars of their language(s)... that could be a whole other question in itself. However, we can state that for complex communication to occur, there should be distinct words, however the linguistic typology will most likely differ from that of humans, and the content would differ due to the different life experiences.
As for communicating with humans, we may well find that the ligers are better able to understand human language than humans could understand their speech, due to their neurology. We can still not understand dolphins or elephants, but they are quite able to understand us in a limited way.
With sociability comes the necessity for social interaction. You can't get away from it without having the blood and corpses of your former pride-mates everywhere. So, we can expect that our ligers would talk amongst each other, and engage in both verbal and physical social grooming. However, due to the lower pitch of their voices, their communication would carry far further than would a human voice in the same circumstances, we could expect that a greater degree of physical separation between members of a pride would be comfortable than would be the case for humans, but they could also be comfortable with sleeping in very close proximity.
As to what would constitute verbal social grooming, for humans it is the weather - we don't like getting rained on much. For ligers, it may be something else of common interest, such as "How's the prey running today?" or "Have you smelled the wind today?"
Humans are highly social omnivorous apes with concealed estrous, and that colours our mindset very strongly. However, Ligers are mildly social carnivorous cats, probably still with overt estrous.
As a top predator throughout their evolutionary history (as opposed to humans who have become top predators relatively recently), we can expect a certain degree of confidence that humans lack. However, even a pride of lions must be wary of other predator species. On the other hand, predators are to a degree cowardly by human standards. A solitary predator cannot afford to take an injury when hunting, so they take as few risks as possible. However, social hunters such as lions can afford to take more risks as they have family to fall back on in case of injury. Ligers would fall between these extremes.
Big cats have been observed to catch the young of prey species alive to give to their own offspring as toys to practice hunting upon. There is no reason why Ligers would be any different. Quite obviously Ligers would see any species incapable of proactively fighting back as being potential prey, but like most predators, they would not be averse to reducing the numbers of other, competing, predatory species if the opportunity presented itself. They would be smart enough to recognise humans as being a predatory species highly inclined to retaliate for the obvious loss of members to predator species, but if the opportunity presented itself for the quiet "disappearance" of a solitary human, they'd jump at it (pun intended).
As predators descending from a species with fewer or more specific mirror neurons, Ligers would have some empathy for their own kind, and to other cats to a lesser degree than humans have for monkeys, but would have pretty much none for other species. Any pain or inconvenience that they may inflict on other species would be either a matter of indifference or utility - if by deliberately causing a non-fatal injury, they could gain some advantage, they wouldn't hesitate. No animal protection society for them.
On the other hand, Ligers may feel quite a degree of like toward other species, much as human farmers tend to like their livestock. That Ligers can like the species that they prey on, and then go on to terrorise and kill members of those species would not be perceived as anything unusual by the ligers.
Ligers, probably having an overt oestrous cycle like other cats, would be unlike humans in their sexual behaviour. Humans use sex as an intimate bonding activity, whereas for Ligers, sex would be an occasional enjoyable diversion and other activities - such as mutual grooming and conversation - would be used for social bonding. Since, like lions, the main social bonds would likely between members of like gender, and mainly between the females of a pride, it is likely that despite any difference in size favouring the males, males would be viewed as the second-class gender, that the females keep around to fight other males and to have fun with when estrus occurs, and the rest of the time, they are considered a bit useless.
Male lions have a tendency to kill the offspring of other males, unless they are closely related, however Liger females may be highly protective of their own offspring, and are unlikely to allow this to happen if they can help it. Infanticide by males would likely be one of those nasty little throwbacks to primitive behaviour that occurs now and then in any species, and the culprit would be unlikely to be allowed to profit from such a crime as may be the case with less intelligent species of cat. Male Ligers would struggle against their instinctive dislike of unrelated cubs and oppose it with the need to be near the female ligers in order to engender their own offspring.
As creatures with some intelligence, however, the female Ligers may allow a degree of infanticide as a means of population control and fitness selection. Very young cubs would be protected as long as they were healthy (and not if they were otherwise) and conditions permit (Lions will abandon their cubs if times are hard), but older cubs may be subjected to tests involving a high degree of risk of death. This could be in some form of military training led by adult males (who would be the primary fighters, as opposed to the females who would be the primary hunters), and it is possible that any significant training injuries or other major failures for cubs would be punishable by death, providing incentive for the others, and strengthening the bloodline by removing defective genetic material.
As a predatory species, Ligers would likely have little sympathy with any members who did not have the potential of being useful members of a pride. Certainly they would look after injured individuals who had the potential for recovery, but crippling injuries, illnesses or birth defects would earn the sufferer only contempt, regardless of that individual's previous strengths.
This section pre-supposes that Ligers do not have genetically-modified opposable thumbs.
With greatly enhanced intelligence comes the ability to manipulate the environment. Cats are not the least dextrous species, despite having thumbs of only limited opposability, as they use their paws to grasp their prey. The structure of the cat paw allows for some limited ability to grasp objects between the pad of the hand and the digits. However, for Ligers, the manipulation of objects is more likely to be a calculated or learned ability, rather than the instinctive activity it is for humans. A Liger wouldn't instinctively think of throwing something, but with the power of observation and reason, they could quickly learn from the example provided by other animals (such as humans). They are unlikely to ever be as good at it as humans, as in comparison their paws are large and clumsy, but what they may lack in finesse, they would make up in power.
A Liger would not see most human doors as much of an obstacle, when they have the size and strength to crash through them, though if they were being stealthy they could use a human door handle and maybe even a round knob (Humans: Recess those circular doorknobs!)
It is possible that after observing humans that Ligers may be able to make simple flint tools, shaped to fit the feline paw. These, in turn, could be used to assist in the production of simple traps, such as covered pits or deadfalls, though a Liger's sharp teeth and claws would be additional tools readily available for most purposes.
Relations with Humans
Ligers would see Humans as inherently weak, making up for serious deficiencies in strength, toughness, claws and dentition with some clever feats of dexterity. On the other hand, a Liger is strong, tough and smart, strong enough that they can make mincemeat of a number of humans in a stand-up fight, and smart and stealthy enough to avoid being shot by those humans' pesky projectiles.
Even if some humans managed to capture a Liger cub - and managed to communicate with it - the cub would inevitably grow to see its human captors as weak and contemptible, regardless of how successfully they might restrain their captive for a while.
For the most part, Humans would be considered as an Other Predatory Species, and no true peace could exist between the two species. Ligers may pretend to compromise with humans, but they would always be looking for an opportunity to rid their range of these competitors. Given the potential danger that a human community poses, any action is likely to be a swift, decisive elimination of that human community in it's entirety rather than a piecemeal reduction in human numbers that would allow and even encourage retaliation.