The last computer that I built was about 3 years ago and its not doing so well. Before you ask; no, I can't salvage any of the parts. So I've been saving up a lot of money for a new $750 budget gaming PC. If you think that you are able to get me a better build than this, be my guest. However, there are some requirements to this build
100+ FPS In minecraft at maxed settings
60+ FPS in skyrim on medium to max settings
60+ FPS in Battlefield 3 on medium to max settings.
The Ability to record minecraft on max settings with a smooth 60+ FPS
The computer has to last a good 5 years.
8GB of RAM
A good enough graphics card to support 512x texture packs
I do more editing than gaming. So I'm not going to get an i3 because once again, I edit. So the i5 will be good for editing and gaming. As for the GPU, I didn't want to get a high end graphics card because usually they cost around $400 and that would eat my budget.
Something like this might be better, the CPU will be better for editing and encoding and the GPU is basically a 7900 series card with less shaders and should be able to max those games with the exception of maybe BF3, it will also help for smoother recording.
The i5 is the better for gaming though, it's up to you really, here are some benchmarks,CPU, 7870XT, 7870 vs 7850.
Also build help topics should be posted in the hardware forum.
OP, use this build it will perform much better your build. Since you want to get high fps in BF3 you will need a lot of gpu power, and the 8320 will perform better than that i5 for $30 less.