[Testing the PSU]
Buy and use a digital multi-meter to monitor the voltages coming from the PSU. Follow the instructions for "Using a multi-meter to check voltages":
http://www.bfgpower.com/troubleshooting.html#DMM
[Testing the Hard drive]
Download the CD image of Hitachi Drive Fitness Test, burn the ISO file to a CD, and then boot from it, just like you would do with the XP/Vista install CD. Test the hard drive and see if any problems are found. DFT will run on most manufacturers' hard drives. Alternatively, you can use Seagate's SeaTools for DOS to test a Seagate or Maxtor drive. For a Western Digital drive, you could use Data Lifeguard Tools for DOS to test a Western Digital drive.
And just for the hell of it:
[Testing the RAM]
Download Memtest86+ v2.11 or whatever the latest version is, unzip it, burn the ISO file to a CD, and then boot from it, just like you would do with the XP/Vista install CD. Let Memtest+ run for at least three hours on
each stick of RAM
separately as well as test the RAM all together. Go for a full 24 hours if you want to be completely sure that the RAM is not a problem. If you start seeing errors, than your RAM is defective or you have incorrect settings for the RAM.
[Testing the CPU]
Use Prime 95, OCCT, Orthos or Intel Burn Tool to stress test the CPU.