For Battlefield 4 multi-player you need a quad core CPU. It is not "blowing half the budget on a CPU". It is making the most from the budget.
Personally I'd go cheap on the CPU and put the money in the GPU:
SteamOS / Ubuntu for Free
$80 Intel Pentium G3258 CPU / AMD Athlon X4 750K or 760K (whatever your brand preference)
$50 1 TB Hard Drive
$50 Basic MicroATX Motherboard (Socket 1150/Socket FM2+)
$150 Geforce GTX 750 Ti / Radeon R7 265 (I recommend Nvidia if you are actually going with Linux)
$70 DDR3 8GB 2x4GB 1866 Mhz Dual Channel Kit
This rig would play most games pretty well.
He already has a 470, buying a 750ti would be pointless.
Haswell Pentium are OK at BF4 are they not ?
