You could build a pretty decent rig with $1000. It also depends on stuff you plan on doing it on it.
If its PURE gaming and nothing much else, then the I5 will be your best bet. However if you do other things such as rendering, streaming and editing the FX8350 might be better for you.
If im honest you wont see much difference real world between 8350 and i5 in gaming.
Good luck.
EDIT: What resolution to you plan to use?