sam9953, this is what I suggest --
Try to use a Linux live CD to see if the system boots up fine and doesn't restart randomly. Also while in this mode try to run tests on your hard-drive for Bad sectors. I think
Dark Star can help you with the nitty-gritties.
Start with
MemTest 86+, download it, burn it on optical media OR put its image on a pen-drive [as stated in the steps on memtest.org] and run atleast 7 passes on your system. This shall ascertain whether the RAM is faulty OR not.
Next escalate to searching for driver OR virus issues by downloading HijackThis and running an instance of the same, it tallies itself with an online database that will help it earmark processes that are behaving irregularly OR have a malicious intent.
And yes please let us know what happens by updating things with a screen shot, that will aid us in helping you out.
How old is your PC? Can we have a the full configuration so that we can ascertain if any hardware is dying on you.
Hope this helps, Cheerio!!