Computer wouldn't turn on... why? How to fix

Hey guys please forgive my ignorance.
I'll try to make this short.

Computer wouldn't turn on, I unplugged the power cord from where it plugs into the green thing... (motherboard?) And now it is working fine, well it's only been a few minutes but it turned right on after unplugging it internally.

When i couldn't get monitor to turn on the fans and disks were spinning, just noting to the monitor.
My question is how to diagnostic the real problem to make sure it doesn't happen again...

More info
It had been acting up for a couple months, turning completely off while it is asleep, or freezing while using it. Then not turning on until I clean the CPU off (but it wasn't really dirty to begin with) or just kept restarting it, many times had to hold power button down.

Computer info
Hp envy
AMD 6120 six core processor
12gb ram
64 bit
Geforce 9800