Madness is a strong word for a strong concept, so not useful in this case, imo. There is no indication he suffered from "mental illness", any more than most young delinquents and criminals do.

What they do share is a sense of alienation from society. He was unpopular, unlike his twin brother, for example. And possibly a tendency to act on impulse when frustrated. 

Again, the facts of the case, i.e. that he almost immediately felt guilty and turned himself in, are telling. That is typical impulsive "acting out". It may be bad impulse control, bad anger management, but not "madness".