In many movies and books vampires have short distance burst superhuman speed. Magic aside what biological changes could be made to the human body to allow it to do something like this?
I would prefer if they would to reach speeds at least 3 to 6 times faster than the normal human max
I have already found biological alterations that can account for most of the vampires powers (strength, regeneration, immortality). But not super speed.
Human muscle can contract 3 times their normal "maximum" if forced to, and if there is nothing stopping them, like overcoming external force. But you already figured out super-strength issues. Speed is similar for the strength in normal humans - up to 3 times normal "max" if you do not need to go fast and internal limiters fail. Both can and will damage other parts of body. Most of people actually could break their own bones or crush cartilages. For your vampires that's apparently no problem, as they can regenerate.
What you need is changes in their neural wiring, to remove limits that keeps people whole. Mother inhumanly fast to catch her baby or daughter lifting car to save her father did happen. Similar level could be normal for your vampires if they can regenerate and their "wiring" accounts for this fact.
To get even faster, it's again up to neurons. In two ways. Faster mammals was found to have thicker nerves wiring their muscle, and more of them. That would be one way to go.
Other, not exclusive, is instinct. People (being homeothermic, aka warm-blooded) have more efficient muscles and muscle nerves than snakes. Snakes are faster. Why? Because they do not think. Simple as that. Their decision making is far simpler. Thus, it takes less time, and it's results are more extreme. One more point to make vampires inhuman.
- Why did you slit her throat?
- Because.
- OK.
Since the question is explicitly tagged "reality check" and asks for "non-magical, biological" changes, the answer must be: "No, none, not possible".
While there exist urban legends of people doing extraordinary feats of strength, those are exaggerated by the press to the point of being untrue. For example, the often quoted teen lifting a car did not lift a car at all. The story is utter bullshit. Lifting a car in this way is something I've done as teen, too, to allow my father creep under the car to check something quickly. While it is somewhat on the heavy side, it does not require superhuman, heroic strength. 90% of the car's weight is supported by the suspension, you only need to lift the equivalent of 20-30 kilograms to "lift the car" far enough so a person can crawl under it (or can crawl out from beneath it).
That aside, the alleged feats of inhumane strength are inconsequential for "vampiric super speed". Gorillas are easily three times as strong as the strongest man, and they don't move anything a vampire. Grasshoppers and fleas are able to move extremely quick, you really have to wonder why. I'll come back to that in a second.
Several things limit a human's ability to move at "vampire speed".
Physics
Did you figure out why fleas are able to move that fast yet? The magic answer is: square-cube-law.
Like everything, our bodies are, too, subject to it. It also explains why elephants, although they are quite strong, can't jump 50 meters.
That aside, your body must obey the laws of motion. Unless magic is involved, it does not have any other choice.
The prime characteristic of "vampire speed" in movies is not that they can generally reach a very high maximum speed (though that is also the case, think of the Master hunting the car in John Carpenter's movie), but that they are able to accelerate (and stop) almost instantly. The light switch scene or the scythe scene in Interviewing the Vampire are examples of that.
In order to do such a thing, one would not need to be a mere three times stronger, but at least some twenty, thirty times (probably more), and your body (including, but not limited to tendons, blood vessels, and bones) would have to be built in a manner to support two-digit g-forces. Accelerating at two-digit-g also includes a few more little details such as your brain banging against the inside of your skull (this is a common injury in car accidents, which is roughly comparable to what moving at vampire speed would mean to your body).
Muscular limits
The speed at which muscles can build up force is limited. Our bodies have several different kinds of muscle fiber (at least 3). The muscles that are strong and enduring are much slower to respond. Which means that there exists a very real limit on how fast you can possibly move, since on the one hand side, you cannot exert enough force with the fast fibers to accelerate your body quick enough, and the fibers that could maybe get somewhat closer to producing the required force are slow responders.
Neurological limits
Standing on two legs, and moving, is an extremely demanding, complex task. You don't usually think about it, but that doesn't mean that it still isn't immensely complex. Ask yourself why robots walking on two legs are still a spectacular rarity.
About 50% of your neurons are found in your cerebellum[1], which is a dedicated organ with no other purpose than to make sure you don't fall to the ground when you walk, and no other bad things happen when you poke your nose.
Some functionality, for example stretch reflexes, is "hardwired" at a lower level with a mere two neurons (well, three, if you count the one going to the opposing muscle). They therefore fire off extremely fast, almost as if you plugged an electric wire onto the muscle.
That is however not at all the case, and cannot possibly be, for a coordinated move, which involves going through a network of a hundred billion neurons. The natural tremor in healthy people (which corresponds to the frequency at which the cerebellum corrects positions) is around 8-12Hz, which suggests that you will not be able to do a lot more than at most 12 distinct moves per second, assuming you need not think about what you are doing.
Walking around a couple of meters (each step being 4-6 distinct movements and three dozen muscles compensating for balance), killing three people, turning on a lightswitch, counting all the grains in that bowl, untieing that knot, and sitting down to smoke a cigarette, all in under a second, and with a bored look on your face... no Sir, sorry.
In order to move at "vampire speed", your neurons would have to be considerably (maybe 30-50 times) faster. Which would mean that oxygen consumption and heat production would go up. As it stands, the brain already is the largest consumer of oxygen (and the greatest producer of heat) in our body.
It is unlikely that you could turn that knob upwards a lot without also taking some very drastic "redesign" measures to the entire body. Since it will be hard to find a better oxygen carrier than hemoglobin, the only solution would be to increase blood flow. That would require higher arterial pressure (and a larger heart), but there's also the problem of limited exchange surface, both inside the lungs and on the blood/brain barrier. Which is not trivial to solve, unless you are willing to increase both chest and head in size.
if you want to avoid tissue damage and run with the theme, you could give them a blood mutation that makes hemoglobin more efficient at delivering oxygen to cells while also carrying lactic acid away faster. this would let the muscles expend more energy more much longer periods of time, and also decrease the possibility of cramping up after anaerobic exercise. not sure how useful the last part would be, but its there.
While the impracticality of Hollywood style "super blur" speed has been addressed, there are some ways for a humanoid creature to have bursts of speed.
This will likely be anerobic motion, utilizing energy stored in the muscles, not provided by the lungs/bloodstream. So it is very short duration and has a recovery time. This can get around the need for a huge spike in blood pressure and aerobic efficiency to fuel muscular contraction. But if the vampire is mostly fast-twitch anaerobic muscle, then he is going to be REALLY sedentary most of the time, as he will have little endurance for extended movement, or at least be fairly weak (languid, perhaps?), since the number of slow twitch muscles in his major muscle groups will be low.
Thicker myelination around nerves conduct impulses faster, as do nerves with fewer synaptic junctions. So the vampire needs a set of "pre-wired" motor nerves that work almost like a reflex. He doesn't have to consciously think about the motion, it is all hard-wired with thick nerves with as few junctions as possible to that the nerve propagation is as fast as possible with out the need to coordinate with slower cognitive nerves. So he can move explosively, almost instantaneously, well before humans around him can react. But he has to do so in a predetermined way, with a narrow range of distance he can cover and actions he can perform (kind of like how a cat will pounce after a laser dot on the wall, shake it off, and give a look that says "did I just do that?").
But what is REALLY probably happening is that the vampire, while really fast, is mostly just PERCEIVED as being fast because he can temporarily "stun" folks. How to do this? Perhaps with an ultrasonic shriek that affects the human brain, causing a very brief pause in conscious thought. Time appears to move slowly during times of stress because we write our memory in high resolution. The shriek, at the edge of the range for our ears to perceive, triggers the reverse. We REDUCE our memory writing, time seems to skip a beat, and that vampire who was across the room is now right in our face. A camera recording the event would see the vamp moving very fast, but not supernaturally so, while all the humans IN THE ROOM would see the vamp just zip across the room while they seemed to be in a daze for a heartbeat or so.
This effect, combined with the near instantaneous reaction time granted by a lot of fast twitch muscles prewired with thick myelin nerves for a reflexive pounce, can account for the ability for a vampire to move in a blur.
Same as with normal animals, only pushed to more extremes. This makes it believable in a plot without further explaining. Animals get tired; sprinting is possible but limited.
The underlying reason is energy storage. Does it deplete local resources faster than it can logistically maintain them?
