Also, whatever you decide to go with, just be aware that you are not going to get a great gaming system for the amount you're spending. The 7850 is a great graphics card, but it's not going to wow you. It can definitely run 1080p but gaming wise you'll get better frame rates if you use a lower resolution. If I were you, and the computer is not a neccessity, I would honestly wait and save up some more so that you can afford atleast the i5-4690k. That processor is probably the best for gaming (and costs about $220), but you won't need to upgrade it for a while and that is key when building a system. If you're going to build a system you'll use everyday (enough to notice its shortcomings), you'll definitely want to make sure to have a great processor and a mid-level card (what you have is fine, especially if you have the 2GB version. You won't get great FPS on high/ultra, but you may be able to catch some decent frames at lower settings.. I'm just the type of person who wants atleast the 3rd best possible gfx card and cpu). Anyways, if you go AMD, you're making a great choice. I'll be honest and say that the i5 2400 is a bit better than the fx 6300, but if you overclock the 6300 to run at turbo full time, you won't notice the difference.. and the stock fan is able to handle 4.1 because that is the turbo frequency. I would pick up a cheap aftermarket cooler if you decide to go any farther than that though.
And yes what Suztera said is correct. You CAN get 60fps, but as I stated earlier, you're not going to get Ultra or High settings which are my preferred settings. If you can handle Medium/Low settings then you can hit 60, but I'd rather play at 30 on Ultra as long as it's stable than 60fps on low settings.
For the mean time I would say just save up, but if you are going to build it now, you should probably get the i3 that was posted above. It won't be as powerful, but it's within your budget and you can save up for upgrades later. Either way, all the best to you! Is this your first build?