This is going to take some serious ground-up modifications to terrestrial animal biology.
In order to have a reaction time fast enough to dodge a bullet, we're going to have to increase the conduction velocity of the nerves from some velocity measured in mere metres per second to something closer to lightspeed. Metal cored and sheathed nerves would fulfill this requirement. Then, we will have to replace the nerves slow diffusion-based signalling mechanism with something much faster... so instead of relying on chemical diffusion, we could have a mechanical connection between nerves. This could be a mechanism in the transmitting neuron which, on receiving an appropriate electrical signal, rotates a certain amount, and which is physically linked to a sodium gate that also relies upon rotation to be opened. Being mechanically connected means that the transmission speed would occur not at the speed of chemical diffusion, but at the speed of sound in the junction rod.
Having reduced the response time of the brain and nerves, the main limiting factor will be the muscles and body. Mammalian muscles are relatively slow. While there are some things that can be done to speed up muscle contraction speed and response time, the fact is that it is unlikely that muscles are going to be able to be made able to contract at a rate much greater than is currently the case. However there are alternatives.
The nature of muscles is that they must contract bit by bit, a few micrometers at a time, but on relaxing, they may be stretched by external forces much more rapidly. So, in order to maximise the velocity at which a limb can bend, we can increase the ratio of joint-to-muscle vs joint-to-load, so that less force is applied, but is applied more quickly. Additionally, in the directions most likely to be needed to dodge, we can replace the muscles entirely and replace it with a highly elastic muscle-ligament combination. In the event of a stressful situation in which it may be necessary to move quickly, the more powerful antagonist muscle would contract, along with the weaker agonist muscle, stretching the elastic ligament. Then, if it is necessary to dodge, the relevant antagonist muscles could be deactivated, resulting in the stored energy in the elastic ligament being applied to the joint much more rapidly than the muscles are capable by themselves. It would also be possible to have both a powerful antagonist muscle and a smaller muscle-elastic ligament combination in each direction of movement, so as to provide two 'gears' to each direction of movement, slow and powerful, and weak but fast.
To complete the perception-reaction loop, we need faster eyes. This is relatively easily achieved, as human eyes are by no means the fastest in the animal kingdom. Even with optical pigments in traditional retinal cells, by making the cells smaller, they could be made to react faster, but with a more radical redesign, it might be possible to substitute a more responsive photosensor more akin to an electro-optical camera sensor.
Finally, this bullet-dodging superman wouldn't likely look like a traditional superman, with bulging muscles and a Mr Universe physique... all that muscle has mass, and the lower the mass, the easier it is to move. Instead, expect a being with long, slender limbs and a slender body, rather more like a grey alien than a human.
However, despite appearing to be slender and fragile, this being would not only be able to dodge with superhuman speed, but could also be an incredibly dangerous martial artist. While its limbs might weigh half as much as an average human's, it would be able to achieve a limb speed perhaps ten times that of a human. Given the relationship between impact energy and mass and speed being E = 1/2MV^2, half the mass equates to half the energy, but ten times the speed equates to a hundred times the energy, for a total impact energy fifty times that of a human's. This slender, lanky, wimpy-looking being could literally demolish a human with a single blow.
Of course, this being's adaptations require that it be aware of the potential attack in order to dodge it. In the event of being threatened, it would crouch, and its muscles would tense up, holding its limbs half-flexed while it stretched its elastic ligaments. It could see a nearby assailant contract his trigger finger, or see the flash of a longer-ranged shot, and within milliseconds, it could deactivate its antagonist muscles, the elastic ligaments contracting to propel it out of the line of fire.
Of course, if caught flat-footed, this being would not have the advantage of having energy stored in its elastic ligaments, and given the likely energy requirements associated with keeping the elastic ligaments stretched, it could not go about with them constantly pre-stretched. In such a case, it would be far more likely that it would be hit by an incoming bullet, though it may be able to achieve a less-serious hit.
Finally, this being, no matter how human it might look, would not be even remotely human. The differences between a human and this being are so great that even if human cells had been genetically engineered to create this being, it is so heavily modified that it would be unable to successfully reproduce with a human partner.