You've already picked the best AMD CPU in that price range.
It's weak compared to Intel's Core i5 but those are significantly more expensive, unless there's a Microcenter in your town (their in-store specials on CPUs are insane- you can get a Core i5 3470 for $150)
Even the Core i3 3240 has approximately the same overall performance as the FX-6300, and is better in some titles (most games still aren't coded to utilize more than 2 cores). But in games which actually utilize 3-4 cores the FX-6300 outperforms the Core i3. So stand pat with your CPU choice.
Instead of synthetic benchmark numbers, let's check actual performance in games:
http://www.xbitlabs.com/articles/cpu/display/fx-8350-8320-6300-4300_6.html#sect0
On the GPU side, the GeForce GTX 650 Ti is the easy pick for $150 or less. It outperforms the Radeon HD 7770 across the board, and gives you the benefit of hardware level PhysX support along with Nvidia's better drivers.
http://www.newegg.com/Product/Product.aspx?Item=N82E16814130838
However, this is a situation where stretching your budget by $5 (actually $10 including shipping) is worth it. The 1GB Radeon HD 7850 for $165 is unbeatable.
http://www.newegg.com/Product/Product.aspx?Item=N82E16814161426
GPU performance:
http://www.techspot.com/review/583-nvidia-geforce-gtx-650-ti/page4.html
http://www.anandtech.com/show/6359/the-nvidia-geforce-gtx-650-ti-review/10
You can safely ignore the fanboy ranting- both AMD and Nvidia have their own proprietary features. Games don't care whether you've got Nvidia's CUDA or AMD's Stream, some titles are more optimized for Nvidia cards (like Starcraft 2, Far Cry 2 and BF3) while others are more optimized for AMD cards (like Skyrim, Metro 2033 and STALKER).
Often the competing cards are so close that it comes down to which game titles you play as to whether an Nvidia card or AMD yields higher fps. This is true if you're stuck between a GTX 660 and HD 7870, or between a GTX 660 Ti and HD 7950.
But not in the $150-$190 range. The GTX 650 Ti is clearly better than the Radeon HD 7770, and the Radeon HD 7850 is clearly better than the GTX 650 Ti. It's just a question of how much you're willing to spend.