Why would my computer start booting then suddenly turn off?

"There can be three things wrong in your case.

Stock fan not correctly fit -- there are some instructions here.

You may have forgotten to connect your fan power to the motherboard.
You did not cleanup the dried out thermal paste properly and the sink and processor surfaces are not connecting properly

Even without any thermal paste layer, a stock sink usually lets a processor run at almost max temperatures (you are discouraged to do this)"

"Be sure that heatsync is properly touching the CPU as well. If you removed it, chances are you disturbed the thermal compound on it and this needs to be cleaned off and replaced.

As a note, you can get a can of compressed air to clean the fan next time. Saves a ton of time and wont damaged your CPU if used correctly."