System will power on runs for a minute and powers off

Your fan speeds up and slows down in response to temperature inside the computer. More so the CPU.

I would suggest getting a can of compressed air made for computers

take it outside, take of the side and blow all the gunk out of there, Really go to town around the fan (you might want to wear eye protection)
While you are in there just check that the CPU cooler itself is not loose

If it is it will have to be re-seated.

If it needs re-seating don't just take it off and put it back on as that might make things worse with your temperature problem they have to be cleaned and new thermal grease put on ($2-5)

DON'T remove the CPU just the fan/cooler assembly.

You must clean both the surfaces without scraping anything off. Use isopropynol alcohol and cotton bud sticks until both surfaces are clean, you can use your finger nail to scrape harder residue if you wish but nothing else.

DON'T pour the alcohol over the CPU

Now apply some new thermal grease (most computer shops have it)

Be careful about how much you put on

Here are some basic instructions

Once you have re-assembled your cooler and re-connected it check everything is OK and power up the computer.

If everything went to plan you should now be back in business