I personally recommend AMD for its prices. If you got money to throw around by all means go with Intel, but AMD does prove cheaper.
The FX-6300 fails between the i3 and i5. i5s are same prices as the FX-8320.
The only reason I mentioned changing the case is because of your low budget, in that case its looks > performance.
What monitor are you going to be looking at? In terms of resolution.
You could get away with a HD 7770 depending on the resolution. In my experience I am able to run Crysis fine at 1440x900, so bare in mind after that the HD 7770 may need some medium graphical settings so not to dip in fps.
I would stay in the 7000 series just because you can get away with a lower power supply wattage