General Corman gets all philosophical (and a little melodramatic, in Shmoop's opinion) on Willard when discussing why Kurtz must be killed:

CORMAN: Well, you see, Willard, in this war, things get confused out there. Power, ideals, the old morality, and practical military necessity. But out there with these natives, it must be a temptation to be God. Because there's a conflict in every human heart, between the rational and irrational, between good and evil. And good does not always triumph. Sometimes, the dark side overcomes what Lincoln called the better angels of our nature.

The Lincoln quote comes from his first Inaugural Address. The movie strongly suggests that Kurtz isn't the only one who's abandoned the "better angels." The entire American leadership in Vietnam has done the same thing. Kurtz is just a more honest version of the same. As Kurtz says at one point:

KURTZ: What do you call it when the assassins accuse the assassin? A lie.

Kurtz believes he's just extending the original logic of the war: subdue and kill the enemy. Corman believes that Kurtz has slipped over the edge into irrational evil. And Willard gets to learn that everyone is capable of having a very dark side.