My mother in law's computer is getting HOT!

My mother in law's computer was working fine yesterday and was shut down properly. This morning, only the Compaq screen would come up without booting the rest of the computer. Then after a few minutes, it would go black...

After turning it off and clearing some dust from inside, I tried again. This time the computer booted like normal, but after another few minutes, shut down on it's own.

I did some testing with the hardware, and found that the CPU was running very hot, although the heatsink is properly in place and the fan working. I cannot figure out what else to try, so here I am.

Should I order a new CPU, or could something else be the culprit?

Thanks for any and all advice!