HELP with gaming PC build for latest games on high or ultra!

For $850? I doupt that you'll be able to get Ultra in Ultra for that, half of that would be needed just for a card to run in ultra (Comparing with BF3) Your best bet would probably get onto Ebay and use your money there, you'll be able to get a better system for your money.

what about this build?
CPU:
AMD Phenom II X4 970 Black Edition 3.5 Ghz
GPU:
NVIDIA GTX 570
RAM:
Kingston 8 GB (2 x 4GB)240 pin DDR3 1333
Mobo:
ASUS AMD AM3 880G HDMI MICRO ATX
PSU:
SeaSonic 620W

That's class!! Hehe, but you'll need a monitor that's full HD to take advantage of games, a cheep case as well. I would also try going for faster RAM if you can push your budget a little?

At minimum I think you need to be looking at GTX 560Ti, 6950, GTX 570 or 6970 in order to play all newer games in all their High-def-Glory.

I'd also recommend avoiding AMD build and try to squeeze an i5-2500k into the build.

Not sure how close to $850 you could make this but somewhere along the lines of:
i5-2500k $225
Z68 motherboard $120
4GB DDR3 1600Mhz CL9 1.5v RAM $30
HDD 7200RPM $80-140
DVD/RW $25
Mid-Tower-ATX case $60
Aftermarket CPU cooler (although not "necessary" right away) $30
GTX 560Ti or 6950 $250
500w PSU (Antec, Corsair, XFX or SeaSonic branded) $80

Prices are just a rough idea, total price: ~$900

For what your wanting to do AMD will be best for you with a budget. Im using a 965BE not even touched its clock and BF3 runs fine with it on maxed settings with my GTX580. Intel works best if you have the money and more for professional uses. But for gaming, cheep AMD does perfect.

As for a case just get any with an intake fan at the front and exhaust at the back, you can get some really cheep ones:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
is more on the lines of all you really need.
Motherboard:

http://www.newegg.com/Product/Product.aspx?Item=N82E168...
Perfect with a black edition CPU:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...