I don't know if this would count as a solid backstory but the thing with Kyle was that he was jealous of the Braxtons. They had their father in their lives, that family, they weren't some dirty little secret, but for Kyle, growing up he didn't have that. He was a dirty little secret, no one knew he existed, not very often did he get the chance to spend time with his father because the Braxtons were his actual family. Kyle only got bits and pieces, here and there, the leftovers. Kyle didn't really know what it was like for the Braxton boys to have Danny around, all he had to go by was his imagination, and I guess he imagined the Danny playing happy families with the Braxtons. He despised them for it, even though it wasn't really their fault.
After Casey got his verdict, the Braxtons were celebrating at Angelos (almost typed Salt lol), and Kyle was sulking. Heath asked what his problem was and he replied "Your brother just got off for killing my dad" (this is a good excuse for me rewatch Kyle stuff on youtube again at 2am). Heath took Kyle to Danny's grave and they talked. Kyle talked about how Danny would take off and he wouldn't see him for months (Heath said that's because Danny had to get back to theirs to kick their butts). Kyle would beg him to stay, and never did get the chance to ask Danny why he always put him (Kyle) second, and Heath that's exactly what he wanted to ask their him.
The main gist of the conversation was that they both felt like they were overlooked by their father, they both were desperate for his acceptance and/or respect. Brax was Brax, Danny saw Brax for Brax, and Casey, he was the baby. The other two were always overlooked and they didn't understand the why. When Casey killed Danny, Kyle felt that he lost the chance at getting answers. So he wanted to inflict some kinda pain onto Casey for taking his dad away from him and taking away that chance at whatever he was expecting from Danny.
You're right, they never really showed us how Kyle went from crazy angry dude to that calmer kinder guy he was when he left. However, I do believe that that particular conversation with Heath that I was referring to above helped put things into perspective for him. That anger he felt towards Casey in the beginning was really supposed to directed at Danny for being that deadbeat father and awful human.
And then he went on to overcompensate and be overly loyal and try really hard to fit in. He never really did fit in though because Brax, Heath, Casey, they were a nuclear unit. It was always them, and then Kyle, their other brother.
Plus he didn't want to be lumped into the River Boy category so he tried to act differently.
I feel that he was misunderstood and lost, not really a bad guy at the core. I love him anyway.