The documentary is the first official word of Kim's role in military operations before his father's death.
It confirms Kim was being groomed as early as 2009 to succeed his late father Kim Jong Il, who died of a heart attack last month.
Among other things, the video shows the younger Kim navigating a tank.
The documentary is the second in a week seeking to highlight Kim Jong Un's experience in leading North Korea's 1.2 million-strong military.
Kim Jong Un, who is in his late 20s, has moved quickly into the role of "supreme leader" of the people, the ruling Workers' Party and the military despite questions abroad about how easily he could assume power with only a few years of grooming behind him.
In contrast, his father had 20 years of training when his father, North Korea founder Kim Il Sung, died of a heart attack in 1994.
Where North Korea is headed under Kim Jong Un is deemed crucial because the country is locked in a long-running standoff over its nuclear ambitions and is grappling with chronic food shortages. North Korea has tested two atomic devices and is believed to be working toward mounting a bomb on a missile capable of reaching the U.S.
After years of acrimony, Pyongyang and Washington had begun discussions about food aid and how to restart nuclear disarmament talks that were suspended when Kim died last month. The U.S. and North Korea fought on opposite sides of the 1950-53 Korean War, and do not have formal diplomatic relations.
The Associated Press contributed to this report.
