The film shows him in his new role as supreme military commander – inspecting troops, saluting and sitting in a tank.
It also confirms that he was being groomed for the role from 2009, showing footage of him from that time watching a rocket launch with his father.
Kim Jong-un’s age is not known but he is believed to be in his late 20s.
The death of Kim Jong-il on 17 December left the threat of a dangerous vacuum in a society raised to venerate one supreme leader, the BBC’s Lucy Williamson reports from Seoul.
Less than a month on, North Korea’s unique media machine is already working to fill it, she adds.
The documentary, aired on state television, shows Kim Jong-un manoeuvring a tank, watching jet fighter and firing exercises and posing for photographs with soldiers.
It also shows him with his father shaking hands with officials at a satellite control centre after scientists launched a rocket in April 2009.
“I had decided to wage a real war if the enemies shot down” the rocket, he is quoted as saying.