|I wonder how many times this question has been asked and answered before?|
Just because your other monitor gets it's power directly from an external outlet does not mean your PC Powersupply is not faulty.
Back to basics.
Do you get any POST beep when you try to powerup? To get a succesful POST you need only Powersupply, CPU with heatsink fan connected, 1 stick ram, graphics (In your case try the onboard)and keyboard.
Take out extra ram, any PCI/AGP cards, disconnect all drives including floppy and CDROMs. Unplug both the power and data cablefs from the drives. Can you get it to POST and display? If not then one or more of the PSU, CPU,Motherboard or RAM is faulty.Unfortunately if it will not post in barebones mode there is no easy way to identify which item is bad. you may need to take it to a tech to test each component. If it will boot then re-connect the other devices one at a time (power off during the connecting process) and restart after each is added until the culprit is identified.
I used to have a signature but it disappeared and I just couldn't be bothered writing another so please feel free to ingore this.