@Name of desire
If you're considering a 570, you have the budget for a real computer without buying some crap one and upgrading on bad components. It's a particularly bad idea to go the AMD route on a motherboard that will probably not allow for much overclocking.
As for SC2 on the laptop in particular, the problem is probably not related to graphics at all. Most likely, your processor isn't up to snuff, or something has changed. For laptops, this is often some a power saving feature being turned on. Power saving features throttle processor speed, the key resource needed by SC2. So look around to see if you've put that setting on or it got tripped somehow. Might clear up your problems.
Quick guide on how to tell if your slowdown problems in SC2 are related to CPU speed: does the game play fine in the beginning, nice and smooth, but as soon as you get to late game and the big battles in particular everything slows to a crawl or even locks up? Textbook CPU problem. The reason being SC2's biggest challenge is calculating all the pathing & AI requirements of the units on the map. So more units = high CPU resources needed. If the problem is graphics related, then the slowdowns will be pretty consistent throughout the game, but that's rare.
With sudden problems in laptops it's almost always a power-saving feature. Though in rare cases there's overheating problems or the player started playing team games regularly instead of single player or 1v1 (more units = more CPU-related slowdows).
As a side note, if SC2 is an important game for you, particularly team games, Intel processors are better than AMD processors (faster per-core).