They will all work with your PC, even nVidia cards.
Which card is right for you depends on how much you are willing to spend and what your expectations are. In terms of performance:
5670 < 6670 < 5770 = 6770 < 6850
The HD 5670 / HD 6670 are the minimum you want to go with on a 1920x1080 resolution monitor, the HD 6670 is roughly 10% faster.
The HD 6850 is a good card that is not very expensive; it can be purchased for as low as $140 after rebate and will allow you to most current games in high quality. If you can live with medium quality graphics in future games, then it should last you for 3 years.
Here's an XFX HD 6850 with a free Deus Ex game coupon for $140 after rebate:http://www.newegg.com/Product/Product.aspx?Item=N82E16814150516