I am trying to build a pc from a motherboard and intel processor that I bought awhile back through the Intel Retail's Edge Program. I put the motherboard (Intel® Desktop Board DG965WH ) and the processor in (Intel® Core™ 2 Duo Processor E6300) in to the case (which I also bought new.) After troubleshooting different problems, I finally got the (heatsink fan) to turn on.However, it seemed that there was no video signal going to the monitor (I have a separate video card, but I was using the built-in VGA output.) and now...sometimes, when I turn it on, the heatsink fan turns on for a couple of seconds and then the fan shuts off. any ideas on what it might be?

Bad connection to mobo from PSU and/or PSU failing and/or already dead.

Yeah dead PSU
Or you cooked the CPU

ohhh..u might not have installed properly the heat sink of u'r processor..try to fix properly the heat sink..if still doesn't work then u'r processor might have some problem..

If the CPU is damaged, there's no fix other than replacement, not usually considered Warranty.