This is a touch over your budget but should be perfect for your usage. You won't be looking at ultra high settings in more demanding games like BF3 and Skyrim but you should get ultra high in most games.
Intel Core i5-3450
8GB Crucial Ballistix Sport 1600Mhz
HIS iSilence HD 7750 1GB
Intel 330 Series 120GB
SeaSonic M12II 520 Modular PSU (Overkill but a steal at $59.99)
Coolermaster Hyper 212 Evo
Total - $724.92 on Newegg
That would give ultra high settings for most games, more demanding ones would probably be more like medium-high and it should be extremely quiet.
If you can't afford that I'd just swap the CPU for the i3-2120. It would still be fine for any game at the moment.
If you don't need a modular PSU, the Antec EA-380D is a cheap alternative to that Seasonic too.