nice budget build. I would change out the 1 x 4gb Ram stick for 2 x 2GB ram sticks so that dual channel memory will be used, otherwise you system will perform poorly.
The XFX HD 6770 will play BF at acceptable frame rates as long as you keep most stuff on high and antialiasing off and anisotropic filtering low x2/x4. you might have to adjust some stuff down to medium though if you experience framing. the CPU is a dual core and will play well on single player, but may suffer a little on some of the 64 player maps, but it is playable.