I was originally going to build a $500 computer, but I've been thinking of saving up a bit more and go to $1000. If I do, would this be nice for a mostly gaming build?
http://pcpartpicker.com/p/16Njf
$500 won't get you very far, with gaming in mind. Your build is very appropriate for gaming. Since this is theoretical, I will just give a little advice. I would recommend a few changes.
http://www.newegg.com/Product/Product.aspx?Item=N82E16817207013&Tpk=xfx%20550&IsVirtualParent=1
I would change the power supply to the 550W XFX. Their power supplies are manufactured by Seasonic. Good quality parts, Japanese caps. Bronze rated for efficiency. You don't need 700W+, 550W is all you need for a fully overclocked system, with a single GPU. The XFX has sufficient supply of amps on the 12v rail for a power hungry card, like the 7970 or GTX 770. Now, corsair do make good power suppies. However, their CX series isn't great.
Second thing I would recommend, is get a 7950. There is virtually no difference between the two cards. You're getting all the performance of the 7970, the same great games bundle, and up to $100 cheaper (in some cases).
http://www.youtube.com/watch?v=7Z6e0UtPZY0
The 7950 really is a brilliant card, great value. Check out the benchmarks in that YouTube video.
Finally, for a $1,000 build, you could probably afford to get a 120GB SSD. It will increase the performance of your OS, and you will boot into games/levels faster. It's a nice boost, helps the system feel optimised.
This?
http://pcpartpicker.com/p/16Uyw