As others have said, save the $ on the CPU and upgrade your GPU/cooler and so on. RAM won't get you a whole lot of gaming performance either, as stated already too.
Anandtech did a test recently, I linked it in another thread and will have to go back and find it. The took every CPU from the i5 2500k (your CPU Latrobe incidentally) right up to the fastest one a gamer would buy, newer socket 2011/3 5930k, about a dozen CPUs in the middle, from 4770k on up to 4790k, and so on. They used a 780ti (fast card 6 months ago when they did the test, not the fastest, but up there). The difference between CPUs in terms of gaming performance with the top 10 games they tested was negligible. We're talking 15% as an absolute MAX, and most were only single digits, from "slowest" cpu to "fastest".
It's very much a GPU world out there in terms of gaming performance, based on that, and other similar articles, and experiences. Based on this, putting your $ if you're limited on what you can spend on upgrading into the GPU instead of any CPU upgrades is probably for the best. IMO the best thing you could do if you can't build a "new" system is keep what you have, fix that cooling issue, and put a 970 card into it. You could probably do this for under 350$ if you look around (coolers aren't that expensive). I doubt your AH, or most any other game performance, would be that much "less" than the majority of newer gaming PCs recently built.