...That may be a problem.
It's starting to sound like your question isn't about a new computer. Because you probably should not be having those problems.
There's something for you to understand. Riot's only given specs is only that you have a 2.0 ghz cpu and a GB of RAM. You see, League is a very simple game and deliberately designed to be so graphically poor that your mom's old, hand-me-down work laptop can run it, because that's the demographic Riot is looking to sell to. League only requires DirectX 9 compatibility, which every graphics card does, and you can probably run it without one.
While you say you have an ATI hd 5670 and the first five seconds of search results show people benchmarking it against Crysis and not getting FPS dips as low as yours when running Far Cry 2.
It's not an issue of you needing a computer, it's that there's something that needs to be fixed.