What are Uncle Axel's character traits in Wyndham's The Chrysalids?

In John Wyndham's The Chrysalids, Uncle Axel is one of the few people in his life that David can trust.

Uncle Axel is seemingly more of a father-figure for David than his own father, Joseph. Where David's father is rigid, inflexible and unforgiving, Uncle Axel seems to genuinely care for David. And it is from Uncle Axle that David gets important—mind-opening, and perhaps some life-saving—information.

David explains that Uncle Axel is "sort of" related:

Uncle Axel was not a real relative. He had married one of my mother's sisters, Elizabeth. He was a sailor then, and she had gone East with him and died in Rigo while he was on the voyage that left him a cripple...he was also my best friend.

Knowing how strict David's father is, it is easy to imagine what pleasure it is for David to have someone who cares about him, rather than someone who worries about "the rules" more than anyone or anything else.

After David is beaten by his father for protecting Sophie, he goes to Uncle Axel with the intention of running away. Axel is able to prepare David with the realities of living outside of the community. When David suggests he could disappear in the Fringes, Axel cautions him that life there is very hard, as is survival. So David asks about places outside of Labrador. Though much of the propaganda indicates that there is no land outside of Labrador that is safe, sailors have seen things and brought back precious metals and spices that introduce questions about these "dangerous places." One of these questions is about Nicholson's Repentances who recorded long dated information that could not be verified. Uncle Axel wonders:

...so you find yourself wondering whether [Nicholson] knew he was in the true image, or whether he only thought he was.

Uncle Axel verbalizes a concern that presents a new perspective for David, which may well be part of what helps him to grow up and get stronger as a person:

I'm telling you...that a lot of people saying that a thing is so, doesn't prove it is so. I'm telling you that nobody, nobody really knows what is the true image. They all think they know...

So Uncle Axel allows David to think "outside the box." Then David describes his telepathic abilities, and his friends—who are the same. Uncle Axel asks questions, even offers to help David find out if something happened to one of his friends. He also strongly cautions his nephew to be very careful. The advice is good because David has never before felt that he and his friends were mutations—or in any danger. Axel warns:

I want you to keep it a 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.

Uncle Axel is a man who knows about the world, so much more than David's father does. He does not judge others: he is very accepting of David and his unusual abilities, and never considers telling anyone what he knows. He worries for David's safety, and give him solid advice to avoid being noticed as unusual about others.

Uncle Axel is a friend—David's best friend—and he's not afraid to ask questions, but he also actively protects his nephew. He has his own views and passes not only his opinions, but a sense (to David) that it is the young man's right to ask questions...carefully.

What is Uncle Axel's role in "The Chrysalids" and how does it influence his response 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.