No, the theory of relativity and all other postulates would still hold true in our universe during the process of being "accelerated" towards the black hole. Remember that deep space is not truly empty, and there will be other particles in the "space" you would be moving through. Considering this, you would encounter a sort of "air" resistance, and this would result in you reaching a terminal velocity which the acceleration due to the black hole's gravity could produce. Since you are travelling theoretically faster than this, and since according to the relativistic equations your mass would be approaching infinity, the added velocity would be negligible if any, and therefore you would continue to travel below the speed of light.
You cannot travel at or faster than the speed of light, because your energy is being converted into mass and so the acceleration becomes so small it is virtually non-existent (and hence negligible). Therefore, this idea is completely impossible. But it was an interesting question...