Computer turns on but does not start up

Hi guys! First off, thanks for taking the time to look at my post. I have what seems to be a fairly unique problem. When turning on my computer everything turns on normally except for the GPU. With the GPU the fans twitch every 5-7 seconds and stop. The thing that makes me entirely confused is that a few times (out of the 30 or so attempts) I was indeed able to boot up and use it just like I normally would. When I did get it up I played CSGO for maybe 15 minutes before it shut off again and would not turn back on. Where should I start to begin this fix or if absolutely necessary what do I need to order to fix this? For what it's worth, the computer doesn't beep at all when I turn it on. When it does decide to work it beeps a few times and then goes through the mobo start screen and the rest of the boot process.

Specs:
-Mobo : Asus M5A97
-GPU : EVGA GTX 960
-PSU : Thermaltake 750W
-RAM : G.Skill Ripjaw 8GB (2x4gb)
-CPU : AMD 8150 Black Edition


Any help is GREATLY appreciated, thanks a bunch guys.

EDIT:

Also, just to catch you guys up to speed on what I've tried :

-Using one stick of a ram, then switching it out for the other stick
-Switching the GPU from one PCI-E to another (alternated RAM each time)

I purchased the GTX 960 assuming my video card was the culprit ~5 months ago. It worked fine up until maybe 2 weeks ago and randomly quit with the same problem as the last card (fans twitching)
As stated in my original post I have (for no reason I could find) gotten it to do a full boot a few different times. Once in Windows I was able to do pretty much anything. I downloaded a program (EVGA Precision) and used it to ensure my fans would kick in at a much lower temperature than usual. It worked for maybe 20 minutes and randomly died again. I can post pictures of how my computer is set up although I don't think that is the issue. I don't have many spare parts laying around to troubleshoot but I'm hopeful one of you gurus out there will be able to help me pinpoint the problem.

EDIT 2: On my PSU there are a total of six ports for PCI-E cables. I only have one (SATA power cable) that runs to both my hard drive and my DVD drive (in that order). Is it possible that I need to purchase a separate cable to run from the PSU to the GPU? Currently it is connected through one that runs off a PCI-E cable from the internals of the PSU. This brings about another question, if it is in fact in need of a new PCI-E cable then why would it have worked in the past and randomly crapped out on me?


Thanks everyone, hopefully together we can figure this thing out!!