Dust the fans/heatsinks.

~2004 was when the Capcitor Plague was in full swing (still is, sort of). Which means ratty power in both your power supply, and your motherboard.

If it's the power supply, that's probably easily fixed as long as you can find a compatible power supply. Which is going to be hard for a circa 2004 alienware, but not impossible.

If it's the motherboard, you're probably c-o-l. (Crap outta luck.) They don't make 'em like they used to, and the only compatible motherboards are either ridiculously expensive, or more crap than the broken one you're replacing.

http://en.wikipedia.org/wiki/Capacitor_plague

Good luck!