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".
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).
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.
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, 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.
It may be small, but it has oomph.