The power supply would worry me. I found another thread kind of going over the power supply that comes with the case. http://www.tomshardware.com/forum/336923-28-will-power-... Normally you shouldn't go with cases that come with a power supply as they are never really all that great. I would suggest seeing if you can't possibly find that case without it and then purchase a PSU from something seasonic based corsair, xfx, antec, and seasonic. In terms of graphic cards the amount of power needed to run a 6870 would lead me to feel that going to a 7850 would be a more beneficial upgrade for your money.
Yea I just wouldn't want to see you fry any hardware by powering up a POS power supply !
Also, yea I the 7850 is a great mid range card at a good price so I think you are pretty solid there. It will blow your 6770 out of the water because technically all the 6770 is is a faster version of the 5770 nothing new in terms of a leap. Also your welcome you'll have a better experience with the 7850 then the 6870 take care!