Why did Azulon want Zuko dead?

Ozai wanted Iroh’s claim to the throne abolished, which made Azulon furious; as punishment, Azulon demanded that Ozai kill Zuko, but the plan backfired.      

Azulon was the leader of the Fire Nation for the majority of the Hundred Year War. His reign as Fire Lord started in 20 AG (after the genocide), until his assassination in 95 AG. He was married to Ilah and had two sons, Iroh and Ozai. He was the grandfather to Prince Lu Ten, Prince Zuko and Princess Azula.

Iroh was set to take the throne after his father stepped down, however, he decided to build a new life in the spirit world instead. Shortly after Iroh’s only son, Lu Ten died, Ozai requested that Iroh be passed over as the heir, which fueled Azulon’s fury. Ozai stated that Iroh was unworthy of the throne, by revealing thoughts of his son’s death to hinder victory at Ba Sing Se. He claimed that the royal bloodline no longer ran through Iroh after his son’s death.

Azulon was furious at the suggestion to let go of his first-born son, and wanted to punish Ozai by making him feel the pain of losing a first-born son. Azulon ordered Ozai to take the life of his first-born, Zuko. Although Ozai intended to take Zuko’s life, his wife, Ursa discovered the plan and plotted a new plan.

The following scene showed Ozai confronting Zuko, stating, “My father, Fire Lord Azulon, had commanded me to do the unthinkable to you, my own son, and I was going to do it. Your mother found out and swore she would protect you at any cost. She knew I wanted the throne and she proposed a plan, a plan in which I would become Fire Lord and your life would be spared.”

Shortly after, Ursa crafted an untraceable color – an  odorless poison that Ozai gave to his father, which caused Azulon to pass away. The following day, Azulon was cremated, while the High Sage crowned Ozai as the new Fire Lord.