Amd fx-8350 or Fx-6300?

An i3 4150 would be my recommendation is place of a 6300. I would recommend either an i5 4460 or 4590 in place of the 8350.

PCPartPicker part list / Price breakdown by merchant

Type Item Price
CPU Intel Core i5-4460 3.2GHz Quad-Core Processor $178.49 @ OutletPC
Motherboard MSI H81M-P33 Micro ATX LGA1150 Motherboard $43.99 @ SuperBiiz
Memory Team Vulcan 8GB (2 x 4GB) DDR3-1600 Memory $52.99 @ Newegg
Storage Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive $44.99 @ Best Buy
Video Card Sapphire Radeon R9 290 4GB Tri-X OC Video Card $264.99 @ Newegg
Case Fractal Design Core 1100 MicroATX Mini Tower Case $24.99 @ NCIX US
Power Supply EVGA 600B 600W 80+ Bronze Certified ATX Power Supply $34.99 @ NCIX US
Prices include shipping, taxes, rebates, and discounts
Total (before mail-in rebates) $675.43
Mail-in rebates -$30.00
Total $645.43
Generated by PCPartPicker 2015-04-05 22:38 EDT-0400

This is just over your budget. Or, well its bit more than just. But, its very powerful. Its capable of 1080p/Ultra/60fps. You could downgrade the GPU to a 280X and save ~$60, but you would lose the Ultra for High in demanding games. However, it can get around 50fps in BF4 on Ultra. You could also downgrade the CPU to an i3, saving ~$100.

/r/pcmasterrace Thread Parent