Hmm, the fact that the computer is working, the way you are suggeting it (though the CPU fan can still spin up when its not working. Check for drives lights coming on, etc)
I would think first, its the video card in some way, since its not giving the signal to the monitor. Open your PC up and take the card out, and push it back in again, re-inserting the video outputs to the monitor too. Does this work?
If its a PCI card, try in a different slot also.
If its an onboard video card, and so isn't any, it could be a chip fault of some sort.
If its a very powerful video cad though, what wattage does your power supply give out? Too low could make it so the card wouldn't power up and if it does, might take a while to warm up to it (i think it could anyway)
I hope some of these suggestions help