If you're comfortable with Linux, you can install a Linux version of StarCraft on your Chromebook:
Keep in mind that this approach requires:
Best for SC2 – free to play.
Tip for SC2: Lower graphics to “Low” or “Medium.” Turn off shadows and reflections.
Playing StarCraft "better" isn't just about software. It's about your body mechanics. StarCraft is an APM (Actions Per Minute) monster.