The FX 8350 is basically the best AMD processor at the moment, there is the FX 9XXX ones but they are just really overclocked 8350's with a different name. That being said the current AM3+ CPU's are 4-5 Years old now.
If you want big performance boost in games, I would be looking at an Intel i5, particularly a 'K' series that can be overclocked. I recently changed from a FX 8350 to a i5 3570k and gained an additional 30 fps on GTA 5, and a large increase in many other titles that I play.
Although if you upgrade to Skylake (6th Gen intel processors) you will not only need a new motherboard but also DDR4 RAM.
I would be looking at a i5 4690k and motherboard to suit. You would be able to keep your exisiting ram too. Depending on your country you might be able to get a new board and CPU for $400.
Or, if you're patient wait until the end of this year when AMD releases their new Zen CPU's and see how they size up against Intel and what their pricing will be. (I personally don't think they will come close to Intel in gaming performance
😛)
Also would you mind telling me the rest of your PC specs
Sorry, I also forgot to even ask: do you mean to increase performance in gaming or just overall computing performance