What is happening as the object is accelerating is that the observer on the object (say, a spaceship), would notice the rest of the universe accelerating and eventually approaching the speed of light, but as far as the observer himself is concerned, he's at rest. It's the same concept of imagining yourself sitting in the back of a covered truck that is driving down the highway: you know the truck is constantly vibrating, but you have no idea how fast you're going, and no way to measure it, or even distinguish it from a truck that isn't moving but is just vibrating. Even if you can look out of a moving vehicle, the only reason that you "know" that you are moving and it's not the world that is moving by you is because you know that the world is so much bigger than you and that it makes sense to think in terms of the world's rest frame, and not in your rest frame. You are, of course, orbiting the Sun along with the rest of the Earth, but how often do you think of that (or notice that) when you're going about your daily activities or just "resting"? The observer on board the spaceship accelerating toward the speed of light observes the same thing: it is the rest of the universe that is moving, not the spaceship.

Meanwhile, what you've read about black holes is wrong. It is the singularity at the center of the black hole that it takes an infinite amount of time to watch something falling into, but which an observer falling in gets there in a finite amount of time. The event horizon is just a distance from the singularity and yes, it is possible to watch something fall through the event horizon of a black hole. Ordinarily, you couldn't see the object falling in that never gets to the singularity, though, because if you're not falling into the hole and are outside of the event horizon, then you can't see inside, and if you're inside the event horizon then you're going to be reaching the singularity yourself pretty soon as well. What you would notice if you were both falling into the black hole is that you would both get to the singularity at the same time, even if the other object had a large head start. What is happening is not that one of you is going back in time, rather that time is passing for you at different rates. An observer watching a clock falling into a black hole would notice the hands on the clock grinding to a halt (assuming the clock could withstand the tidal forces of falling into the hole, of course). An observer falling into a black hole would also notice that time in the rest of the universe would become increasingly rapid as he fell in, and as the observer reached the singularity; time in the rest of the universe would be passing infinitely quickly.