You could interpret a dream about being chased by a clown to mean that you're running away from or avoiding problems from your childhood, or that your (or somebody else's) immaturity is causing you trouble.
Dreams About Being Chased
A dream about being chased could mean that you're running away from a problem in your life, or that there's a difficulty that you're afraid to face and overcome. Think about the thing chasing you - does it relate to any aspect of your life that you're having difficulty with?
Dreams About Clowns
Clowns that appear in dreams are often believed to symbolise the playful, light-hearted, childish aspects of your personality. However, if you have a phobia of clowns, or if the clown is scary, this could represent an insincere or deceptive person in your life. This doesn't mean that you should suspect your friends of being untrustworthy, but it could mean that you're afraid of being tricked or lied to.