Hello wakinews,
In my experience it's most likely the PSU.
Also check the CPU FAN to make sure it is working.
A CPU rarely dies unless it has overheated a lot, or it's over volted.
Make sure the PC is clean from dust build up - fans, heat-sinks, vents, etc.
If you have a lot of devices attached, you could try removing them and see if it helps.
If the PSU is dying that might lower the power draw enough to boot up...
The best way to test is to try known working components.
i.e. borrow a good PSU and test your PC with that, or test your PSU in another working PC.
That's not always easy, possible, or practical...
If you know a reliable repair shop you could see how much they would charge for testing to determine the cause.
They should have equipment/parts that they can test with.
Personally, i would buy a new PSU and try it.
Even if that is not the problem in this case, it never hurts to have a spare back-up PSU on hand.
You do use a good surge protector or UPS, right?
hope this helps,
David
