Combining AMD's budget processor with a solid graphics card will net you some good overall FPS in games especially if you're willing to tweak the settings. Even if you're not, then you'll still be able to play games like Battlefield 4 on ultra (2 x AA) with a framerate above 30FPS.
The Better CPU Choice:
If you'd rather sacrifice some FPS for a better CPU later, then go for the FX 6300 along with something like the HD 6670. It certainly won't give you great FPS on high settings, but it will allow you to play most games in low to medium settings in 1080p without a problem.
Upping it $50 for an Under $250 Budget:
If you really wanted to get a little more FPS you'd up the price point a bit and reach for the new GTX 750TI. Overall, it wouldn't give you a lot more performance in a game like Battlefield 4, which is optimized for AMD, but it would give you a significant 10-25% improvement in other games.
That being said if it's me my first priority would go to the CPU. Switching from a processor like the 760k to the 6 core FX 6300 seems like the better move overall.
