That's a low-profile card so it should fit in the slimline case; there should be no issue with the board either. You'll have to remove the full-length bracket and VGA cable and attach the included low-profile bracket.
Assuming your power supply is at least 240w or 250w, then you should have no problems running a Radeon HD 6570. However, getting a 2GB version is a waste of money. Video cards do not use more than 1GB of VRAM unless you play games above 1920x1200 resolution. The Radeon HD 6570 is not powerful to even handle playing games at 1920x1200 resolution unless you set graphics quality to low. Even then the performance will be rather lacking.
You are better off simply buying the following 1GB version for $40 after rebate. You will not see any performance difference between 1GB and 2GB of VRAM.