There's really no way to justify spending your $237 CPU + Mobo + Cooling budget on an FX processor unfortunately. I don't have time to kit out your build for you, hopefully someone else will come along to do that, but you could save a few dollars dropping to an i3 which is at least equivalent and usually significantly faster for gaming. Or up your budget by about $20 and you're on a Skylake i5-6500 with a decent entry level mobo which will decimate the FX in games while using a fraction of the power. If the $1K is a hard limit, drop to 8GB RAM and you have the cash for the 6500.
One other point... spending $210 on a last-gen GPU right now is a real shame. The RX 480 is a much, much better card which should drop to $220ish for decent aftermarket 4GB versions once demand settles (prices are stupid at the moment). If you absolutely have to buy now then it's probably an okay call, but if there's any way you can hold off for a few weeks to see if prices settle, you'd get a much better long term rig.