An 8320 or 8350 should run fine on your motherboard. Be wary of anyone who tells you to buy a 6300 instead. It's a decent processor (I'm using one now), but it isn't quite up to snuff when it comes to the more hardware intense games that are on their way-- if you can afford an 8000 series, get it, along with an aftermarket fan and heatsink (the 8320/50 is HOT).
Of course, you're going to need a new GPU too. Even a 270x would be a pretty big leap from what you have now.
If you can't afford both a new GPU and CPU, buy a new GPU first. The Athlon II x4 640 will work relatively well on newer games, but pretty soon the 560 will useless.
Note: You may need to buy a new Power Supply for all of this. Please be careful not to overload your computer!