Time to dissassemble a little and see if you can put a dab of conductive heatsink compound on the CPU and put it all back together. It sounds like heat is the issue or you have the original battery and the batteries need changing (BOTH CMOS and the big one.)
Just guessing, but cheaper than a new laptop.
Here's why you may have NO CHOICE but to replace the main battery. As it ages (and fails) the cells can short out causing a load on the laptop power systems. I've seen people ignore this battery and run on AC and after a year or two the laptop power supply fails. Too bad they didn't get a new battery.
I have a Gateway Solo 5300 laptop (PIII 600mhz, 192 MB ram, 10 GB HDD, Win XP Pro) and it won't stay on when I try to turn it on. Sometimes I can get it past the login screen but normally not even past the BIOS screen. I have determined that this is not a driver problem due to it not even showing the BIOS screen before shutting off. The CPU fan runs all the time, all the dust is cleared out of the intake and exhaust. When in windows when it shuts down first the screen goes black then I hear static coming from the speakers then it shuts down 5 seconds later.
Yesterday I thought I had this fixed when I reset the CPU and replaced the thermal compound. It ran for 15 hours straight, I then shut it down for a few hours today then this afternoon I tried to turn it on and thus back to the drawing board. I will note as well that the battery only lasts 15-20 minutes. But I have been using the external power adapter this whole time.