Going to also agree that the GTX960 will decimate the R7-260x. Flat out no competition.
As for the new CPU, what are your parameters? Need to keep the same motherboard? Willing to replace motherboard and CPU both? To be completely straight, once you hit a certain level of CPU, it won't choke the GPU except in very certain circumstances on certain games, and the difference between a top level CPU (i5-4670) and a mid-low level CPU A8-7600 is only a few FPS when paired with a mid-level/high-level GPU.
That minimum CPU level BTW, to play most games at 1080p with very high/ultra details is the aforementioned a8-7600. Your A6-6400 unfortunately WILL bottleneck most GPUs and be relatively unplayable at any decent level of details/resolution.
The good news is that you have quite a large variety of CPU's that you can drop right into your motherboard which will make things MORE than playable, and what I would do, since you're going to have a dedicated GPU is just get a good, solid Athlon x4 860k. $75.00 or so at Newegg. Unlocked, overclockable, and more than powerful enough to not choke your GTX960 for a minimum amount of cash.