In TFA I would say he was a very good villain, raised in a well intentioned environment and part of the Skywalker bloodline, he had everything going for him. His cathartic devotion to a dogma past simply because of the ideal of Vader turns him into the whiny petulant child with a lust for power that Anakin often was. There was a set up for him to be Rey’s foil, pushing himself to his physical limits to surpass hers, for the sake of none but his own. Murdering Han was not so much killing his past, rather cutting the emotional tethers that would withhold him from his peak performance. Generally a once good hearted kid who is now forcing himself to become a sociopath because of a pre ordianed destiny that he unwittingly concieved by himself.
TLJ complicates this description, as has been discussed to death by this point. There is less concern with upholding the legacy of Granddad and instead a more independent definition of reaching self-actualization. Kylo’s mission is now said to be breaking the cycle of war, and taking its precedent out of the Galaxy. At first, so we are left to assume, he hopes to achieve this through destroying the Resistance…which would just leave a proxied Empire in power, essentially still the same deal as thirty years beforehand.
His purpose in the movie doesn’t really make sense until Snoke rejects him, and at that point he then has a motivation to want to rid the world of both Empire and Rebellion alike. It then falls apart again when he appoints himself Emperor and once again returns to the idea of just taking out the Resistance.
By the end of the second film, he is reduced to screaming and a desire to kill, not really moving in any particular direction. I feel as though the scope of the character has now become so vast and confusing that we would need a defined conclusion in part three for the whole thing to once again make sense.