Every character needs a good home.
An important aspect of designing a successful character involves giving your creation a sense of solidity and grounding: it needs a backstory and purpose. I find that including these in your thought process will significantly improve your character design.
The second part of this extended project, which started here, explores how to use different aspects of your character's personality to create a solid and relevant environment for it to exist within. Ask yourself: where does it live? What does it do? Once you've got the basics nailed, you're ready to create an environment for your character, and a stunning piece of artwork for your portfolio.