Im thinking you have XP?
Whatever you have, try starting in safe mode, maybe remove some of the stuff you put on it to see if that helps with the BSOD. (the stuff you really don't need anyway.)
It could also be a failing hard drive, memory, or other hardware device. (Try removing video card if it also has on board video, this will test that component.)
If you decide to get a new computer you can always go with a cheaper refurbished one, but of course the pc may or may not run the games you really want. (Mame games are usually a safe bet, other emuls maybe not.)
But with the Pentium processor you have now, going up in processor model will usually run what you are running now.
You can also re-install windows and keep all your programs, if you have the disc. Just select that option when you boot from the disc. (Complete reformat may not be necessary.)
Edit: just back up all your stuff if you can, because reformat may be your only choice if you have ruled out hardware problems.