Since you get nothing on the screen, my guess is that the problem is power supply or motherboard. That is "good" news. You can remove the drive and attach it to another computer to rescue the files on it.
You didn't say if this was a laptop or desktop. If a desktop, you generally just pop the drive out and pop it in to another desktop. If a laptop, it is more difficult--both getting it out and attaching to another computer. The way to go here is to get an inexpensive USB-attached hard drive case (just make sure it is the right size for the drive-desktops are 3.5in and laptops are 2.5in in general). Note you can do this for a desktop drive as well, but you don't have to.
Here is a page of examples.
As sad as it is for a computer to die, the most important thing is the data. It sounds like in this case, the data should be fine.
Best of luck,
WallyH