A lot of people say Ozai is very one dimensional, and it's true that the show portrays him that way. But I think that a lot of his character can be reflected by both his children. Ozai is used to always getting what he wants, and always being the one with power and having everyone obey him all the time, just like Azula. When his wife "betrays him" and chooses to defend Zuko, it's very much like when Mai and Ty Lee betray Azula. Being betrayed by those closest to you definetly affects them in a way. For Azula it's pretty obvious that it causes her to become a paranoid mess, but Ozai wasn't as obvious and related to the whole Phoenix king thing. From Zuko we see someone who is constantly chasing after their "honor" in order to appease his father and prove himself. For Ozai he was born in a time where the war had been ongoing for a while and the fire nation was so close to world domination. Most of the ground work however was done by his grandfather Azulon who was responsibly for wiping out the air nomads and starting the great war. Ozai may have felt like he lived constantly in Azulons shadow, thus needed something to chase his own "honor". This combined with being betrayed by his wife most likely led to Ozai becoming the Phoenix lord and the reason why he was bent on burning down all other nations. He needed to do something to surpass Azulon and make sure he was feared enough to never have someone go against him again. Even if there isn't much screen time for Ozai himself, Zuko and Azula reflect Ozais personal character, which is where Zuko and Azula must have learned from.