If that price for the 5970 is correct, then that one easily.
If you actually mean $699 then best value would be 2 stock 5850's OC'd. That's about the cheapest that'll actually run that resolution well. It'll win a FPS/$ comparison (but cheap cards will always win, since $/FPS goes up practically exponentially)
1080p gaming with 2 monitors will require more than 2 5770's, so that's out.
2 5850's can do it if you OC. $580
1 5970 should be able to handle it. $700
2 5870's will give you better performance. $800
Overall, factory overclocked cards aren't worth it, since you can do the same with a few clicks in ATI overdrive.
I think the only games you won't be able to max and get "as good as it can be seen performance" are Crysis and Metro 2033. Those will struggle a little with the AA and AF turned up. If you want the best possible graphics performance (which sounds like what he wants), in a single card solution, go with the 5970. Beating that is HD5870 xfire, but now we're talking 800...
Also, with the 5970: It is two 5870 chips on a single board downclocked to 5850 speeds. The heatsink is a 400 watt capacity one (someone correct me if I'm wrong), so it's got plenty of overclocking headroom. If you overclock your 5970 cores, you should be able to get the performance of two 5870s.
Dollars per frame (DX11 , 1920 x 1200, hi settings, 9 game average) costs
ATI 5970 11.51
nVidi 480 10.01
nVidia 470 9.08
ATI 5870 10.59
ATI 5850 8.60
Here's another comparison of the higher cost board solutions under $700 ... twin 5850's look the most attractive here price wise as far as min. frame rates .... although the 470 looks better at avg frame rates. Metro 2033 is the only "spoiler" in the bunch putting enough load on the 5850's and 5970 under the 30 fps minimum....with a bit of OC tho, the twin 5850's will climb that methinks.