Well... I think I had the same problem with my computer a few months ago.. The fan on my graphics card decided to die on me and the reaction I got from the computer was it starting up normally except the monitor staying completely dead.
I could hear the whole thing loading etc and the fans turning in my computer but the monitor would remain dead even when I replaced it with another.
Now best thing you can do like Gimbly said is open the thing up and have a look inside it.. and maybe keep the thing hooked up and turn it on whole you got it open so you can see if the little fan on the graphics card is turning.
Because if its not the computer simply cuts off the power to it to make sure it won't overheat so that explained the black screen.