$1000 is a great budget if it's only the PC and not screen, keyboard and mouse too. I know the Radeon HD 7950 is an amazing card for overclocking and you can get up to 140% of the performance if you overclock it. This makes it probably the best card on the market, as an overclocked 7950 is just a notch better than the GTX 680 in most games.
As you can see from this graph, the 7950 is a great card but not close to the best ones.
Now, look at the average FPS when it's overclocked to 1200/6000 Mhz
Of course, the 7950 definitely isn't cheap, probably twice the cost of a 560 Ti, so it might not fit your budget. Just thought this information might be useful.
Btw, the graphs are partially in Swedish but it says "Higher is better (FPS)" and green is the average and red the lowest fps.