What is the role of Uncle Axel from The Chrysalids? How does that role affect how he responds to the Purity Laws?

Uncle Axel is David's friend.  Among the adults, Uncle Axel is David's best friend.  The friendship is deeper that simply an uncle and nephew that like to spend time together.  Uncle Axel is a trusted confidant of David's too.  Uncle Axel is the one person that David trusts to tell his secret to.  David knows about the Purity Laws, and he knows that being a deviant is considered a bad thing.  That means that David knows he needs to keep his telepathic abilities a secret, but David does tell Uncle Axel about them.  It's a huge risk because Uncle Axel could immediately tell Joseph Strorm about it, and David's life would be over.  That is not what Uncle Axel does though.  He immediately grasps the importance of keeping the secret, and he works to help David and the other telepaths keep their gifts a secret.  

"I want you to keep it secret. I want you to promise that you will never, never tell anyone else what you have just told me — never. It's very important: later on you'll understand better how important it is. You mustn't do anything that would even let anyone guess about it. Will you promise me that?"

It's at this moment in the story that Uncle Axel moves from being a friend and confidant to being a protector.  At that moment, Uncle Axel is also agreeing to be a Waknuk law breaker.  He is required to report deviant genetics.  Instead, Uncle Axel keeps the secret, and that puts him at risk too.  Uncle Axel takes his self-sacrificing protection to drastic action when he kills Alan.  Uncle Axel kills Alan because Alan found out about the telepathic abilities, and Uncle Axel figures that dead men tell no tales.  Basically, Uncle Axel's love for David allows him to completely ignore any legal obligations that the Purity Laws have placed on him.