The jump from a athlon to a phenom is huge aswell as a 6850 to a 7770 so if you can try and upgrade both, another thing is try and get memory that is faster 4gb is enough for light gaming but 1333 is slow for anything so concider that aswell
I would go with a new CPU, if you have the budget I would switch to one of Intel's sandy bridge perhaps the core i5 2500k that would give you a good boost, in pretty much everything you do on your pc!
You really need both upgrades, especially for Battlefield 3. You won't be able to max the game out with a 6850 or 7770 either, but I would still suggest going with the GPU first, since games are generally more dependent on the GPU than the CPU.
I just saw this video in my subscriptions. It might help you make your decision:
Think of it in these terms, your CPU is dealing with the physics going on when it comes to dealing with vehicles, they have weight, speed, health, can colide are affected by gravity and the slope they are on too some extent. Every time they fire your CPU has too deal with the transversal between you and the object that was fired upon and that changes if speed, distance or angle are changed and that is constantly being processed and updated, and is being updated for every Vehicle, RPG, Missile, Explosive or pretty much 'thing' that moves or kills people dead.
I hate budget builds. Just save, save and save. that way you'll be happy with what you bought and you won't need to go out and buy the parts you should have in the first place