For gaming, particularly first person shooters, buy the best graphics card your budget and PSU will allow.
Since your prices will differ, use the heirarchy list at the end to rank your different options.
If your TV was 1080P, it is likely that you will still be gaming at the same resolution, namely 1920 x 1080.
Since your 5570 is not that strong of a card, I have no doubt that your budget will buy you a significant upgrade.
I'll offer my opinion. From my experience assembling budget to mid-range gaming PCs, for 1920x1080 gaming, stay with a HD 6870 or greater. Or a GTX 560 or higher if you prefer Nvidia. The HD 7870 seems to be a hot item right now. But I'm not sure if that might not be above your budget you set.
What about manufacturers?
Like i've been looking at the MSI r6870 With the dual fans and single fans and powercolor versions -
I may wait for the GTX660 though and see how it fairs out;
I've been told by many people that the 6870's can compete without overclocking with the 560 and 560 Ti ;
And i can add to the budget it was just an estimate
The graphics card marketplace is very competitive. You will get fair value for your dollar at every price point.
Cards in the same price category will have similar performance.
One exception is that older cards which are less effective may still retain high retail prices. The reason, I think, is that they can be sold to sli/cf upgraders who have nowhere else to go.
For the most part, the basic card and components all come from amd or nvidia. The various vendors apply their stickers on them and add custom coolers for differentiation. It is mostly fluff.
Newer 28nm cards do run cooler and use less power so I think that is the way to go.
Some vendors are US based if that makes a difference to you. It might if you need warranty service or other support.
Check out the various forums.
Warranty and rma policies may differ too.