Alright, first off lets clear this common misconception. Having logged a whopping 222hrs across multiple computers I have found that BF4 is not a cpu extensive game at all. Almost all quad core cpus can run this game max 1080p with like 80fps.
Also take a look.
http://www.techspot.com/review/734-battlefield-4-benchmarks/page6.html
The Athlon x4 750k is not on that graph but it is better than the FX-4100 by a longshot.
To max the game out you will need a beefy gpu though, that's why I've recommended the GTX 660 because it a solid performer and fits the price/performance ratio almost perfectly.
I run the game max with a weaker cpu than I've recommended.
The new athlon server cpu's from amd have shown to be quite cost effective.
I use Hitachi drives because they have the lowest failure rate.
An mATX board would better fit this budget.
Besides guys, I got a GTX 660 in there most games are going to benefit from a better gpu.
Check her out
CPU: AMD Athlon X4 750K 3.4GHz Quad-Core Processor ($79.98 @ OutletPC)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($29.98 @ OutletPC)
Motherboard: Gigabyte GA-F2A88XM-D3H Micro ATX FM2+ Motherboard ($69.99 @ Newegg)
Memory: Corsair Vengeance 8GB (2 x 4GB) DDR3-1600 Memory ($69.99 @ Newegg)
Storage: Hitachi Deskstar 1TB 3.5" 7200RPM Internal Hard Drive ($66.75 @ Amazon)
Video Card: EVGA GeForce GTX 660 2GB Video Card ($189.99 @ Newegg)
Case: Fractal Design Core 1000 USB 3.0 MicroATX Mid Tower Case ($29.99 @ Newegg)
Power Supply: Corsair CX 600W 80+ Bronze Certified Semi-Modular ATX Power Supply ($54.99 @ Microcenter)
Total: $574.66
(Prices include shipping, taxes, and discounts when available.)
If you already have a hard drive and compatible case let me know.
Great case might I add I've built quite a few builds with it and I've heard nothing but positive reviews from customers.
-Auto merchant update link-
http://pcpartpicker.com/p/2SruS
Remember this build recommendation is a work in progress. I will be updating it as parts change prices. Need any help PM me on the forums and I'd be glad to help!
Much love
-Cappy