Learn French: C'est moi qui me touche le menton avec le genou ! - It is me who touches my chin with my knee!

This is the standard French way to talk about actions performed on your own body. Instead of using a possessive adjective like mon, you use an object pronoun (here, the reflexive me) to show whose body is affected, and a definite article (here, le) before the body part. Literally, you are saying "who touches to myself the chin".

Why is it avec le genou and not avec mon genou?

The reflexive pronoun me earlier in the sentence already establishes that this entire action is happening to and with your own body. Once that context is set, you don't need possessives anymore; you just keep using definite articles (le, la, les) for any other body parts involved in the same action.

Why are the verb and pronoun in the first person (me touche) if the subject is qui?

In the phrase C'est moi qui..., the relative pronoun qui represents moi (me/I). Because the antecedent is the first person, the verb must be conjugated in the first-person singular (touche), and the reflexive pronoun must match it (me).

Could I just say Je me touche le menton... instead of using C'est moi qui...?