Ozai is Zuko's father, although Ozai's and Zuko's definition of "father" varies greatly from one another. From Zuko's point of view, the most human thing his father ever did was save him from drowning when he was three. There were supposedly times after that when his father was "good," or at least when Zuko saw him as a good father. What Zuko knows for sure is that the desire to be Fire Lord corrupted his father—or potentially corrupted his father further. It was commonplace for Ozai to pit Zuko and his sister Azula against each other. Despite Zuko being Ozai's firstborn, as well as the eventual Crown Prince after his cousin's death, Ozai preferred Azula much more over Zuko. Still, he tolerated Zuko and, up until a certain point, didn't mind Zuko's misbehavings, such as his unwarranted presence in the war room when Zuko was thirteen.
It was that war meeting that severed the thread between Zuko and Ozai. Zuko, in a bout of compassion and justice, insulted the general, scolding him for being so willing to put freshly recruited soldiers in a battle he knew they would lose simply for a preemptive strike. However, Ozai took the insult personally. Zuko was ordered to fight an Agni Kai—a fire duel—as punishment for his disrespect. Zuko accepted without hesitation... at least until he saw who his opponent was. Unable to face his father out of love and respect, Zuko acted shamefully in Ozai's eyes by bowing and refusing to fight. When threats didn't work, Ozai finished the Agni Kai and burned a third of his son's face off. He then went beyond that punishment by banishing Zuko. However, he was "merciful" and claimed that if Zuko found the Avatar, he would be allowed back home.
And so Zuko went on his quest to find the Avatar—a journey that was supposed to be impossible.
Although it's very clear that through most of the series Zuko wants to be in the good graces of his father, there are times when he acts childishly, calling his father foolish in the presence of high-ranking officials such as Zhao. In contrast, when he is alone with Iroh after he's been given the opportunity to go home without the Avatar, Zuko turns against his uncle harshly, claiming that Iroh has been nothing short of selfish and jealous of his brother Ozai as if Iroh hadn't just gave up three years of his life to accompany Zuko. And yet, once Zuko commits the ultimate betrayal against his uncle and gains his father's good will, Zuko remains unhappy. After a while, Zuko realized that all he ever wanted was for his father to love and accept him the way he was—as the man who wasn't perfect; someone who didn't want to continue the war and who wanted peace. When Zuko confessed these feelings to Ozai during the Day of Black Sun, Ozai retaliated. He manipulated Zuko into staying until the sun came out by bringing up his mother and the fact that he knew what happened to her. Once the sun came out and firebending was possible, Ozai firmly told his son that banishment was too merciful a penalty for treason and that Zuko's fate ought to be much steeper. Ozai then generates lightning and, aiming to kill his son, shoots it at Zuko. Thanks to Iroh's lightning redirection training, Zuko was able to redirect it and escape unscathed.
After Team Avatar defeated Ozai and he was stripped of his bending, Zuko had no problem putting his father in jail. Zuko seemed almost bitter about the fact Aang left Ozai alive and eventually, that decision came to affect Zuko. When his line of questioning about where his mother was failed, Ozai taunted Zuko saying that being Fire Lord was tough—that Zuko would crumble under the pressure. And crumble Zuko did. Within a year, Zuko went to his father for advice because it seemed everything was falling apart. He even brought tea!
Although Zuko acts like he doesn't care about his father, it's clear he does. His hope about whether or not Ozai will start acting like a father fluctuates often because he remembers the time Ozai was a good father. Advice from people like Aang and Iroh doesn't help. When Ozai arrived on the Thor on November 11, 2012 he was ironically rooming with Rapunzel. Already fearing terrible things were going to happen, Zuko convinced Rapunzel to move in with him just to be safe. While Zuko leaves Ozai alone for the most part, he's still nosy and tries to see what his father might be up to and whether or not he has his bending.