iMac Shut Off - Won't Turn Back On (Fully)

So, last night I was using my last-iteration iMac and it randomly shut off. I tried to power it back on... and it wouldn't. I then felt the back of the case and it was very hot. I figured it was an overheating issue, but now it won't turn back on. When I hold down the power button, it begins to start up and I can hear it "clicking" then it shuts down. It restarts itself, and I can hear a very low hum. Something is running inside of it, and the disc drive will spin if I put a disc in, but it's not turning on fully. I tried several different troubleshooting techniques with the main problem being nothing will appear on the monitor so a few of them are out of reach. I also tried booting to the boot disc and, again, nothing will appear on the screen. I did the one power reset which caused my fans to blow loudly and I had to, again, fix that, but... yeah. It starts up, shuts itself off, starts back up and something works but I can't tell what is whirring in there. The computer is only about six months old, but I didn't take out the Apple Care so I'm extremely frustrated, as it is well-cared for and this issue seems extremely random. Thanks in advance to anyone that can provide help.

Message was edited by: Jonathan Glover