yeah, you could build a decent gaming rig for under 1000$, since I do this I'll give you a couple of tips as to what you might want to spend and skimp on to get the bang for your buck.
1). 3 things to put the money into
1. Power supply - ABSOLUTELY do not cheap out on this part, spend at least around 100$ because if your power supply goes, chances are the rest of your computer will, I sell computers for a living and it amazes me the people that will build a nice rig and cheap out on the PSU and then have to buy everything over again once their system breaks because it fried the motherboard, GPU, and CPU, and RAM. I like antec, corsair(if you can spend the money), and seasonic(if you can spend the money, they are the manufacture for corsair PSU's)
2. Mother board - Spend a decent amount on this, around 100$-150$ will give you a good mother board with a good array of ports and expansion slots to upgrade you're computer easily
(gigabyte being a personal fave, I like MSI as well)
3. GPU, choose whether you want an amd or NVidia and try to get the best bang for the buck, so try to spend around 250$, go for 300$ if you want a better one, but after that point you will start wasting money on getting marginal gains in performance...the gtx 660 and 670 are good buys for NVidia and the AMD Radeon 7870 and 7950 are good buys for amd. EVGA is by far my favorite card manufacture for NVidia as their customer service is phenomenal. For AMD I like sapphire, and MSI are pretty good too as they do a better job than other manufactures at the heat sinks.
2) Things that are OK to cheap out on:
1) CPU - if you are only doing gaming and nothing too intensive like photoshop or CAD, an i5 would be fine and an i3 would work too. Most games don't really use multiple cores/threads of if they do they use like 2. so this is one area where it won't make much of a difference.
2) RAM - I would go for like 8Gb, anymore won't make much of a difference with gaming and you could possibly get a way with 4Gb, but since RAM is cheap, just do 8
3) Case- unless you want to keep the case forever or you want one that's super stylish, a cheap one will do the job just fine.
4) dvd drive-not really necessary anymore, buy a cheap lite-on and save the money
3) iffy- HDD, you can cheap out on but the higher end ones do last longer IMHO so I would recommend spending the money if you can.
4)SSD- I recommend these for every computer. If you can spend the money, buy at least a 64gb ssd for the boot drive as it will do more for the speed of your computer than anything else can. The HDD is the slowest part on a computer so an SSD makes a huge difference on any machine.
that being said, you can definitely build a good system for around 1000$ maybe 1300$.