Does your cpu heatsink fank come on immediately and run well?
I'd verify that it is installed correctly, and properly mounted on the CPU (sometimes it's difficult to push it down enough so that it'll be tight against the CPU).
When a computer turns off after a small amount of time, I'd suspect an overheating problem forcing it to turn off to be safe. If you can get into the bios, can you get to the screen where the temperatures are?
In some cases, the motherboard detects the CPU incorrectly and sets it to the wrong speed, which will quickly cause overheating and make the machine power off. I'd verify that it's being detected correctly.
You should also check over the board, make absolutely certain that your case wires are connected correctly to the power and reset on the motherboard. I'd start disconnecting other components like the hard drive just to verify as well.
You should look over the motherboard to make sure that there are no blown capacitors (they'd look like the top or sides exploded if so).