Black Holes cannot currently be directly observed, since they swallow up the light and radiation we would use to see them. It would be possible to detect distortions where light is bent by some unseen object of great mass. Determining whether such a distortion was from a genuine black hole, as opposed to distortion caused by a Massive Compact Halo Object or other hypothetical form of Dark Matter would probably be difficult and dangerous. As such, a Black Hole might serve as a navigational hazard in a role-playing game.
In most settings, we can assume there will be some way of detecting and avoiding a black hole. Established space lanes and commonly traveled routes will avoid black holes and suspected black holes. The ships' computer might analyze visual data from the sensors, compare them to star charts, and compute likely gravitational distortions in the vicinity. With superscience or applied phlebotinum you might be able to build a sensor that accurately detects the warping of spacetime that occurs near a black hole.
So, black holes should really only be a danger to vessels exploring remote areas of space, or to ships that for whatever reason have lost their navigational sensors, computers, or star charts. For the sake of drama, though, we'll assume you've somehow bumbled upon a black hole without warning. It's not going to look like a big area of darkness as you might expect, at least not until you get very close. Instead, light from objects far beyond it will be bent, so you'll actually see behind the black hole. Objects on the other side of it will be distorted, but if they're just distant stars the distortions might not be immediately obvious. At near ranges to the event horizon, you'd be able to notice the big black spot, but the relativistic speeds used by most fictional space propulsion systems may well mean you have very little time to react.
At the Event Horizon, light is in orbit. As you cross that horizon, if you look perpendicular to your motion, you'll see light that has circled around. If the Schwarzschild Radius is small enough, you may actually see yourself in the distance, or even a chain of yourselves, as light runs circuits around similar to what happens on a hypersphere. Of course, if the black hole is large, the diameter of the event horizon may be so long that you and your spacecraft are tiny specks. One dramatic element the GM could narrate in would be the ship's automated proximity alarm going off, as the computer suddenly detects several other ships. This phenomena would alert you to the fact that you'd reached the Event Horizon, but it'd be too late to escape unless you have an FTL Stardrive. Unless you can travel Faster Than Light, the best you could do at this point is establish an orbit of the black hole. Again, the likely speed your craft was traveling at, and the relatively narrow band of the event horizon, means you're not likely to be in the orbital plane for long.
As your orbit decays and you fall into the black hole, things start getting distorted. This is going to be a nightmare, but at least you won't be staring a multiples of yourself any more. You'll be stretched in the direction of the black hole, and squished in the directions perpendicular to that. The parts of you closer to the singularity will feel much greater gravity, and start falling inward faster than the parts of you that are further from it. This stretching is called "spaghettification", but it'll actually look more like what happens when you stretch a broken rubber band. As you get longer, your midsection gets thinner.
Objects closer to, or further from, the singularity would get dimmer, and the light coming from those directions would have a reddish tone. This is the result of red shift, the phenomenon of the light taking longer to reach you. Looking perpendicular to your velocity, everything would be brighter, and have a slight bluish tone. This is the result of blue shift, the phenomenon of light taking less time to reach you. Your view would become distorted to include more of the plane perpendicular to the radius of the black hole, and our binocular vision would have serious trouble interpreting the distorted space. It would appear that the universe had been flattened into a plane perpendicular to your motion. These distortions would eventually become so strong that you would die, and then be crushed into the singularity. You'd be ripped apart on one axis, and crushed on the others.
Time would dilated as you accelerate into the black hole, but you'd be unable to notice. If your spacecraft were very large, you might be able to notice things ahead of you slowing down and stretching. It probably depends on the size of your ship, the size of the black hole, and the speed of your approach.
An outside observer watching your approach on the Schwarzschild Radius would see a very interesting display, though. First, you'd start to fade - this is the redshift effect again, as light coming from you takes longer to travel the distance to them. You'd appear to stretch. Your travel toward the center of the black hole would speed up, but the dilation of time would mess up their perception of it, and your other motions would seem to slow down. Eventually, as you cross the event horizon, you'd vanish from sight completely, as the light bouncing off of you would get stuck on that side of the Event Horizon.
Caveat: I do not have any scientific credentials, so it's certainly possible that I may have misunderstood something in the sources I read. It's not like your players can say "that's not what happened the last time I fell into a black hole" though, so it's probably accurate enough for gaming. At least, accurate enough if your players aren't professional astrophysicists.
3. Non-Fiction Book: Hyperspace by Michio Kaku
Read more about Black Holes at Arcana Wiki.
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