Failing RAM sounds like a good place to start here.
Head to Memtest86+ - Advanced Memory Diagnostic Tool
and download MemTest86+. Run that program for roughly 5-10 passes. This tool is going to check your RAM to see if there are any bad pieces in it.
If it passes, try to load in Safe Mode with a command prompt and run "chkdsk /r". This will scan the entire surface of your hard drive for errors that may or may not be fixable.
These are the first two areas I would check and in your case the two areas that are most likely to be causing issues.
If both come back clean, please let us know and we can proceed from there.