Very tough choice.
The FX-6300 is about $100.
The FX-8350 is about $170.
The i5-4440 is about $185.
The i5-4690K is about $230
There's a huge difference between how well certain games run as AMD CPU's tend to bottleneck many games. Having said that on a fixed budget you have less money for the graphics card. So it comes down what is the best BALANCE averaged over the games you'd tend to play.
My advice based on current pricing is to get the FX-6300 plus GTX770 or consider an AMD graphics card of similar value which may work better in newer Mantle supported games like Sniper Elite III (included in the three games of choice in the Gold Tier for certain AMD cards).
We are likely going to see more of a demand than I used to think for having MORE than 2GB of VRAM at this performance level.
Thus, if we assume about $350 for the CPU + Graphics budget my advice is:
a) FX-6300 +
b) R9-280X 3GB
Not sure why the average user score at PCpartpicker differs so much from NCIX scores but you can research that more:
http://pcpartpicker.com/part/asus-video-card-r9280xdc2t3gd5
Summary:
There are PROS and CONS to either way but my advice for roughly $350 gets an FX-6300, R9-280X 3GB, and three free games.
Here's a link to the Asus R9-280X (about same as GTX770):
http://www.techpowerup.com/reviews/ASUS/R9_280X_Direct_Cu_II_TOP/26.html