1. Put it in another pc.
2. Check the voltages etc with a mutimeter, not an easy job and I can't tell you how.
3. Open it up (carefully) and see if you can see anything wrong like bulging/leaking capacitors, burnt bits.
Personally I'd find another pc to test it and/or open it up. If it is the psu then it's quite likely that it's blown capacitors.
Quite a cheap fix, if you do it yourself anyway (never paid anyone to fix em myself).
Plenty of info on replacing cap on this site and here's a link that covers everything, read it before you try
opening it or changing any caps
Bad capsCould also be a motherboard or processor issue, and i'd say that is more likely, but the psu could be at fault..