Gravity is sometimes described as a curvature in space-time. Due to relativity, doesn't this imply that gravity doesn't propagate?
There's a fairly precise sense in which gravity propagates: if you have a spacetime and you perturb it a bit, then you can think of the new spacetime as the old "background" spacetime with a small change on top of it. Then it makes sense to discuss the speed at which this change propagates.
But in the general case, the speed of propagation of gravity has no particular meaning. This is sensible: to talk about speed, you need some standard to compare it to, and hence a background spacetime. But gravity is nonlinear, so to have an objective answer to that question, the changes to that background need to be either small or tightly constrained.
If a black hole was moving toward you at the speed of light ... [would you] experience something similar to a supersonic blast, except its a gravity blast, ...
Not analogously to a supersonic (so here, superluminal?) one, no.
For an electric charge, the ultra-relativistic limit of its electromagnetic field is a plane wave that's impulsive, an infinitely-thin Dirac delta profile. This travels at the speed of light. The gravitational analogue of this (for a Schwarzschild black hole) is the Aichelburg-Sexl ultraboost, which is a spacetime that's axially symmetric and everywhere flat except for an impulsive plane wave.
... and it would allow you to remain unaffected until you arrived at the very centre of the black hole?
If you were able to turn around, just before you arrive at the center, and travel away from the black hole at the speed of light you would never notice it, even though you would be travelling at the same velocity as the black hole in the same direction?
That's the same thing as saying you're stationary above the black hole. Hence, you would notice it because you'd need to be accelerated to stay stationary. Plus, if you're turning around after passing the horizon, it's far too late.