Im pretty sure BF4 or any of the CODs are compatible with linux, even if you use Wine. Also, here is a build that is just a tad over budget, but if you can wait for a rebate, it fits.
PCPartPicker part list / Price breakdown by merchant / Benchmarks
CPU: AMD Athlon X4 750K 3.4GHz Quad-Core Processor ($79.99 @ Amazon)
CPU Cooler: Cooler Master Seidon 120V 86.2 CFM Liquid CPU Cooler ($49.99 @ Newegg)
Motherboard: Gigabyte GA-F2A88XN-WIFI Mini ITX FM2+ Motherboard ...