How do I clean the bottom of my climbing shoes?

Depends what you have on it. If you got paint on it, and it doesn't come off with water or by rubbing it off, then you could try putting alcohol or something similar on it, otherwise, you could use a soft wired brush to remove it.

The bottom of climbing shoes are usually just rubber, so keep that in mind when you use something to clean it, make sure it can be used on rubber and better to test it in a small place on the shoe that can't ruin it, just in case.