When in close proximity to a black hole does the gravitational influence speed you up to the speed of light?

No mass can reach the speed of light. No matter how much force is exerted on it. So stuff, mass, will speed up as it's sucked in by the extreme gravity. But it'll never reach c, light speed.

But light, photons, are already there at the speed of light. That's why it's called the speed of light.

And photons do get attracted by the extremely warped space that the energy of black holes causes. So photons are sucked into a black hole and there they stay because they lack the requisite escape velocity to get back out of that hole. That is, c, the speed of light is not sufficient to escape the energy well inside the event horizon of any black hole. That's why they are black...the light does not get back out.

This question makes me think that you think that a black hole speeds things up to the speed of light. I think this is a misunderstanding.

Objects with rest mass never reach the speed of light. To accelerate them to that speed would require an infinite amount of energy, which not even a black hole can provide. Massless objects (such as photons) always travel at the speed of light, regardless of where they are.

The key about a black hole is that the gravitational effects warp space such that even light, traveling at the speed of light, cannot escape through the event horizon.

As with all questions of velocity and relativity, you always have to ask relative to what?

If you are in flat space far from a black hole, you will see an object at the event horizon appear to stop in relation to your reference frame. An object's velocity appears to approach zero as it approaches the event horizon.

If you are an object approaching the event horizon, you can consider yourself at rest in a Lorentz frame. One of the interesting things about general relativity is that in a suffiently local region in any strength gravitational,field, you can always find a 'locally flat' space-time in which special relativity applies. Everything seems normal.

If you are in your super spaceship firing your rockets so that you are at rest with respect to the event horizon, you can see an object approaching the speed of light but never obtaining it as it crosses the event horizon.

So, there is no single answer to your question. It is all relative.