Try running the bare minimum, disconnect all drives and their cables from the mobo, remove extra add-on cards like soundcards, video editting cards etc. and external peripherals. The only thing you need to test is: 1 ramstick, a videocard, keyboard, mouse and monitor.
Try ram-sticks one-by-one, and another power supply and videocard if you can get your hands on those (Performance doesn't matter, as long as it gives you a display-signal). Try another monitor even. And even disconnect your case-wires if all else fails (reset, powerbutton, usb-ports. You can jumpstart your pc by shorting the powerpins with a screwdriver, check the manual).
If you're lucky to regain a display-signal you start adding ram-modules and exchange your own parts one-by-one again until it fails again. You should be able to pinpoint the problem this way.
You gonna have to turn on/off many times, so it's better to press the normal powerbutton 4 seconds (or short the pins 4 seconds) rather than using the hard powerbutton on the back.
Also check your mobo if it has any led-indicators that might indicate what's wrong (check mobo manual). Some mobo's also give beeping-signals to indicate what part is failing.
Also, listen closely to your mobo if you can hear any hissing that might be caused by faulty capacitors. Replacing capacitors is difficult but not impossible in (roughly estimated) 50% of the cases. (although modern mobo's use less and less traditional capacitors).
It's possible of course, but CPU's rarely die, especially nowadays as they all have failsafes for overheating.
