In 1967, during the Vietnam War, Army Special Forces Colonel Walter E. Kurtz (Marlon Brando) has become insane and now commands his own Montagnard troops, inside neutral Cambodia, as a demi-god. Colonel Lucas (Harrison Ford) and General Corman (G.D. Spradlin), who are growing increasingly concerned with Kurtz's renegade operations, assign U.S. Army Captain and Studies and Observations Group veteran Benjamin L. Willard (Martin Sheen) to terminate the Colonel's command with extreme prejudice.
Willard, while ambivalent about the mission, joins a Navy PBR (for "Patrol boat, riverine") commanded by "Chief" (Albert Hall) and crewmen Lance (Sam Bottoms), "Chef" (Frederic Forrest) and "(Mr.) Clean" (Laurence Fishburne) to head upriver. They rendezvous with surfing enthusiast Lieutenant Colonel Bill Kilgore (Robert Duvall), a commander of a cavalry squadron, to discuss going up the Nung River. Kilgore initially scoffs at them, but befriends Lance when he discovers he is an expert surfer and agrees to escort them through the Viet Cong-held coastal mouth of the river where surfing conditions are particularly good. At dawn the helicopter raid commences. Amid the attack, Kilgore calls in a napalm sortie on the local cadres and the rivermouth is taken. Willard gathers his men to the PBR, which has been transported via helicopter, and begins the journey up river.
Tension arises between Chief and Willard as Willard believes himself to be in command of the PBR while Chief prioritizes other objectives over Willard's secret mission. Slowly making their way upriver, Willard reveals part of his mission to the Chief to assuage the Chief's concerns about why his mission should take precedence. As night falls, the PBR reaches the chaotic last US outpost on the NĂšng River, the Do Long Bridge. Willard and Lance enter the base seeking information on what is upriver. Unable to find anyone in command of the base, Willard orders the Chief to continue upriver as an unseen enemy launches a strike on the bridge.
The next day, Willard learns from a dispatch that another Studies and Observation Group (SOG) operative, Captain Colby (Scott Glenn), who was sent on an earlier mission identical to Willard's, had joined Kurtz. (A few days before Willard received this dispatch, Chief had told him that about six months prior to Willard's mission Chief had taken another man north of the Do Long Bridge. Chief had heard this man shot himself in the head.) Meanwhile, as the rest of the crew read letters from home, Lance pops open a purple smoke grenade for fun. It attracts the attention of an unseen enemy in the trees, and Mr. Clean is killed during the firefight. Further up the river, they are attacked by the natives once moreâone enemy throws a spear that impales Chief, and in his final moments, he attempts to kill Willard by pulling him down onto the spear's point. Willard retaliates by suffocating him. While Lance buries Chief's body in the river, Willard finally reveals his mission to Chefâdespite Chef's initial outrage at the fact that Willard is going "to kill one of our own guys," he rejects Willard's offer to let the two of them head back downriver while Willard goes on alone, and insists that they complete the mission together.
The PBR arrives at Kurtz's outpost, and the surviving crew members are met by an American freelance photojournalist (Dennis Hopper), who manically praises Kurtz's genius. As they wander through the compound they come across Colby, who stands nearly catatonic, along with other US servicemen now serving in Kurtz's renegade army. After returning to the PBR, Willard later takes Lance with him, leaving Chef behind with orders to call in an airstrike on Kurtz's compound if they do not return.
In the camp, Willard is subdued, bound and brought before Kurtz in a darkened temple. Tortured and imprisoned, Willard screams as Kurtz drops Chef's severed head into his lap. After several days, Willard is released and given the freedom of the compound. Kurtz lectures him on his theories of war, humanity and civilization while praising the ruthlessness and dedication of the Viet Cong. Kurtz discusses his family and asks that Willard tell his son about him in the event of his death.
That night, as the Montagnards ceremonially slaughter a water buffalo, Willard stealthily enters Kurtz's chamber as Kurtz is making a tape recording and attacks him with a machete. Lying mortally wounded on the ground, Kurtz, with his dying breath, whispers "...The horror... the horror...". All in the compound now sense something amiss in Kurtz's quarters; seeing Willard departing the room, dropping a bloody machete and carrying a collection of Kurtz's writings, they drop their own weapons and bow down, allowing Willard to take Lance by the hand and lead him to the boat. The two of them motor away, as Kurtz's final words echo eerily for the audience, as their world fades to black.

Question: Why did tension arise between Willard and Chief?
Answer: Tension arose because Willard thought he was in command of the PBR while Chief prioritized other objectives over Willard's secret mission.
Alternative answer: They both believe they are in charge.