First thing, Welcome to Bleeping Computer.
Now on to your problem.
When you say, "When I run the Hardware Diagnostic tool" are you referring to the diagnostic tool that is on the Windows Vista DVD?
I've never put much stock in that test if that is what you are using. I would first recommend running Memtest86+
and checking the memory in your system. I would also recommend possibly cleaning the interior of your system to cut down some of the heat you may be experiencing. Heat can cause many different problems including your BSOD. Click here
for a basic tutorial on cleaning the interior of your computer along with the normal precautions for doing such thing.
If after cleaning your computer and running Memtest86+ your test results come back as normal, please try running your computer as normal and let us know what happens.
One more thing you may want to look for when cleaning your computer is any capacitors that look like this.
If you have any capacitors exhibiting these signs, or having the centers bulged up, you will probably have to have either the defective capacitors replaced or if you are unable to find someone to do that job, a replacement motherboard will be your next best thing.
Hope this helps,