http://anandtech.com/bench/CPU/1107
Your Phenom is going to be just a bit behind the FX-8350 in there...and since the the top i7 on that list paired with a GTX 770 (like your GPU) only outperforms by a bare 3 FPS at 1080p, I really doubt you'll see significant improvement.
Nor should you get an FX CPU, unless your Phenom is running into physical issues. The FXs (8320, 8350, etc.) are only 1 tier above your Phenom, & Tom's doesn't recommend upgrading for a single-tier upgrade.
Yes, going with an i5 (4590, 4670 or 4690, with or without the "K") would be a 2-tier upgrade. But, you have to balance that against having to not only buy the processor but also a new motherboard...and therefore having to reinstall Windows from scratch. Even if you can get to a local Micro Center & get a bundle, you're running at least $250-300 USD...and, at least for BF4, that's a lot for a very minor uptick in performance. Same with GPU; your 770 is near the top (5th-tier), & you'd have to get an R295 X2 or a GTX 980 to meet the "2-tier-minimum" upgrade recommendation.
If your CPU is giving you lots of problems or is on its last legs, your cheapest bet would be to get an FX-6300. It would be brand-new (thereby lasting longer), performs slighlty better than your Phenom (http://anandtech.com/bench/product/699?vs=102), & Tom's currently gives it Honorable Mention as an entry-level gaming CPU (http://www.tomshardware.com/reviews/gaming-cpu-review-overclock,3106-2.html) due to its low price. Whether or not you do that, though, you can start saving up to plan for either a Skylake upgrade, or to see if AMD releases anything later this year/early next year.