amd fx 6300

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yes it will help in the overall performance and you have to think to that most games out right now are singlethreaded meaning they only use 1 process core per thread and most games cant use above a four cores any way there are a few that make use of hyper threading and the way 8 core visheras divide tasks. a 8 core vishera is good and computes faster than a 6 core but a 6 core is fine as well. i still would be reluctant to put a 780 with it because of the speed of the card. a 770 i think would be the limit but with your setup a 4gb 770 and a nice overclock on the 6300 should do you fine. but the longer you save up the better performance you will have so its up to you.
 
and a d-14 can handle some heat. that is pretty much the best air cooler you can get. here is some benches for a 770
http://www.techpowerup.com/reviews/ASUS/GeForce_GTX_770_Direct_Cu_II_OC/10.html
as you can see your resolution will be some where in between the 1080p and 1600p benches if you cut them in half on bioshock infinite you should be getting somewhere in between 80-90 fps on max settings. with your cpu it may look more like 70- 80fps just a guesstimate. probably being a little harsh but just so you can get an idea. and i assume the panels are all 60hz so you will be fine. i mean games like crysis 3 sure your not going to max that but even a 780 wouldnt max it.