you may have one of the earliest dual cores on your hands, but you may be being lied to you (with a clever technology that Intel uses).
either way; spending money on that computer is money wasted.
i don't know how much money you have currently, or when you'll depart with it so take the following with a grain of salt. this is what i would recommend to a gamer-on-a-budget; it's got plently of grunt at a cheap price and is moderately upgradable in the future.
CPU: AMD Phenom X3 720,
GPU: any HD4830 (don't get me wrong, i love my nVidia card to death, but ATI's current crop of cards are superior to all but the really high-end nVidia cards),
RAM: 4GB of DDR2-800 (as cheap as you can get it)
Mobo: anything with a 790GX chipset.
you can recycle your old case and hardrives, you may need a new power supply, but ask your local computer geek about that as it's really something that should be done in person.
just remember; this could become out of date at any moment (and if a random person is reading this in a months time, it could be out of date right now). be sure to do your own research and fact-checking, it won't be my fault if you don't.