My path would be cpu/cooler/psu (small OC) - GPU - new mobo if needed.
The problem is that although a 6300 is going to be an improvement, in BF4 the 8320 is going to be a far better choice, and they're too close in price to dump one and upgrade to the other. And of course the 8320 will just throttle in that board (if it even supports it).
Maybe the 8320s will be on weekly offer tomorrow - that'd help.
I'd be more inclined to go new board now, 7850 on offer/b-grade, new CPU later.