I have a bit of a speculative idea about this. I think that Kylo, growing up, was probably the type of child who was prone to anger, frustration, and emotional outbursts, to a greater extent than most others are. Perhaps he couldn't shake these tendencies, so that's why his parents sent him off to train with Luke (and in retrospect, a pretty *****ty reason to send your kid off to "holy warrior school"). I'd imagine that Luke explains the differences between the Jedi and the Dark Side, (as well as Vader's story) and Ben, in his mind putting 2 and 2 together, assumes that his personality type fits in with the dark side moreso than it does the Jedi. Snoke potentially whispering in his ear may have exacerbated this as well. From there, I can easily see how a twisted rationalization could lead to him wanting to carry on Vader's legacy. In effect, he's not a true believer in the sense that he's wholly indoctrinated by Sith/Dark Side ideology, but he's doing what he does because he feels that this is where he's supposed to fit in.