I don't think it's a heat issue, heat would most likely just cause the CPU to issue a HALT command (causing everything to freeze in place).
If it boots into Windows (or at least gets past the Windows XP booting screen into the welcome screen), then you might have the Sasser worm. Microsoft has a page that gives a little information at http://www.microsoft.com/security/incident/sasser.mspx
and you might want to search Google for more information about how to detect it and how to remove it if you do have it.
If you do in fact have the Sasser worm, then that means you were not keeping your computer updated. You need to either turn on Automatic Updates or visit www.windowsupdate.com
every so often (Microsoft generally releases a bunch of updates on the second Tuesday of every month, so that is the best time to check, but it doesn't hurt to check once a week or so). It is VERY important to keep your computer updated, because the new big thing is "Zero Day exploits" which is when hackers grab the latest Microsoft updates the day (and sometimes the hour) they come out, take them apart and examine what hole they patch, and then write a virus to take advantage of that hole. Then, they release that virus into the wild and anyone who doesn't get updates regularly will be vulnerable.
Frankly, I don't think the Geek Squad will be much help, although I don't have much faith in them in the first place - I think they overprice for things that should be common knowledge or at least can be found on Google. It certainly is possible that you have a hardware issue, but I think it is more likely that you have some sort of software conflict if not the Sasser worm (or other virus/spyware). If you still have a warranty, you might want to consult Dell, but I am thinking you don't have a special warranty because you consulted Geek Squad first... Warranties are your friend...
If you do not have the Sasser worm, let me know and I'll see if I can come up with any more advice for you (or WackyT can beat me to it).