If it weren't for the conflict between Claggart and Billy, there would be no reason to tell the story of Billy Budd. Billy would be a nice admirable sailor who just so happens to have been drafted on board the H.M.S. Bellipotent. Many of the first chapters deal more with the setting than with the characters, providing a sense of the historical backdrop to our story, a sense of the extent to which the fear of mutiny is alive and well in the British fleet. Because we don't get any backstory on either Billy or Claggart, the initial situation is as much a historical scene as it is the narration of Billy Budd's story. It is as if the narrator must build himself a stage before he lets his play be carried out upon it.
We don't know the exact moment where the conflict begins. The reason is that the initial conflict is not a question of action. It has to do with an inexplicable phenomenon: Claggart's hatred of Billy. The narrator speculates that Claggart is naturally depraved (given to bad or evil acts). If there's an ounce of truth to this, one could even argue that the conflict began long before the H.M.S Bellipotent, that there has been some evil in Claggart that has been seeking expression for a long time.
Once we know of Claggart's hatred, we begin to observe the little ways that it manifests itself: Claggart makes sure Billy gets in trouble for little things around the ship; he has to hide his anger when Billy spills his soup; he hires an underling to spy on Billy; and he probably talks an afterguardsman into trying to implicate Billy in a mutinous plot. But all of these actions stem from Claggart's animosity toward Billy, an animosity whose origin is unknown. At times, the narrator can't help but jump out of his own story and try to come up with explanations for Claggart's mean streak. The key point, though, is that it's a mystery.
At first, all of Claggart's aggression toward Billy is passive, though there is a feeling of fatefulness about all of it. If only for the narrator constantly telling us so, we know things are going to get worse. Things don't actually get complicated, though, until Claggart approaches Captain Vere and makes his accusation. You might even trace the complication to a single line. It is the moment when Vere asks Claggart to identify the mutinous sailor and he says, "William Budd, a foretopman, your honor" (18.14). The accusation is so serious that, as soon as Claggart mouths the words, we know we have enough conflict for the rest of the book. As Captain Vere makes clear, either the accused or the (false) accuser is going to be hanged.
A clear argument could be made that the actual climax of the story is Billy's execution. After all, that would seem to be the moment when Billy's fate is decided, the moment when Vere sees the consequences of his decision. Yet we would like to argue that Billy's fate is decided long before the noose is around his neck, even before the meeting of the drumhead court. Billy's fate is decided the moment his fist makes contact with Claggart's forehead and kills him. Just before, there is a period of unbearable tension when Billy is struggling against his stutter, trying to work up some words to defend himself. The catharsis comes with Billy's act of violence, and at least as the story is presented by the narrator, from that moment on things simply run their course. Captain Vere, upon realizing that Claggart is dead, cries, "Struck dead by an angel of God! Yet the angel must hang!" (19.13). At this point he knows what will happen, and since the decision lies in his hands, we do too.
In some ways, as we've pointed out, after Billy hits Claggart, the rest of the story is just "going through the motions." We know what's going to happen. The suspense, though, results from the predicament of Captain Vere. We feel his struggle as he attempts to reconcile his idea of himself as a good man with his role as the Captain of a ship. A moral dilemma has been placed squarely on his shoulders. The result is that the story stays incredibly taut right up until the moment of Billy's execution. Since the narrator has already told us what is going to happen, we realize that the only reason we're still in "suspense" is because we're sympathetic with Captain Vere. Given that Billy is going to be executed, we're asking: Is Captain Vere going to get out of it OK? Or is he going to lose his mind?
After Billy's been hanged, the tension more or less goes out of the story. From this moment on, we see characters reflecting on what has already happened. There is the key moment where a murmur goes up in the crowd, and the boatswain silences it by calling everyone to duty. Later, there is the purser who imagines that it was Billy's will power that kept him from twitching. The surgeon dismisses the idea, and wants him to regard it as natural chance. The surgeon, like all of the characters in positions of power on the ship, is dedicated to making sure that Billy does not become a symbol. The fear is that even though he never had any mutinous intent in his life, his unjust death could incite mutiny all the same. The denouement, then, is the quelling of any mutinous thoughts in the crew.
The story closes with three distinct memories of Billy Budd. First, we get Captain Vere, who murmurs Billy's name over and over again as he dies. Second, we get the totally inaccurate article in the paper that depicts Billy as a traitor who stabbed Claggart out of vindictiveness. Third, we get a poorly written sentimental sailor's poem commemorating Billy, and we find that in some circles Billy has become a mythic figure. Clever as we are, we realize that the story that we have been told is really just one more version of what happened. As we realize this, our attention again becomes fixed on Billy Budd, on the idea that he represents, and on who the man actually might have been. The conclusion, then, force us to looking backward over the events as they were told; it is a dictum to re-think everything we thought we knew.