Are you
Only planning on playing LOL and some web surfing ? Your PC is definitely not good for gaming. You can't just add a Graphics Card to improve performance , like what the others have said , more RAM , and a new PSU would definitely improve your gameplay.
There won't be a lot of games you could play with the new upgrades. You could try call of duty and some free online games.
the resolution of the monitor you are planning to play on also matters. the higher the Resolution , the lower the FPS.
The HD 6570 1GB GDDR5 Requires
22 amps on the 12v rail and a max wattage output of 400 watts
The HD 6570 2GB DDR3 Requires
18 amps on the 12v rail and a max wattage output of 400 watts
The HD 5570 1Gb DDR3 Requires
24 amps on the 12v rail and a max wattage output of 400 watts
Source:
http://www.game-debate.com/hardware/?gid=415&graphics=Radeon%20HD%205570%201024MB
You should upgrade everything if you really want to see an improvement.
I could recommend to you some cheap parts , but you would have to tell me where you live and what your budget is.