Billy is the main character of Slaughterhouse-Five, but he's not exactly the hero of the book. Or rather, he doesn't have the heroic qualities usually associated with the main soldier in a story about wartime. In a word—okay, three words—he's a weirdo. Who wears silver boots.
Billy is a funny-looking optometry student when he gets drafted to enter the military. He is sent to fight in Luxembourg against the Germans in the Battle of the Bulge and is promptly captured by a small group of German scouts. He manages to get on nearly everyone's bad side by being so feckless and pathetic, with his constant nightmares and his weak body, but he does survive where a lot of others don't. He even manages to make it through one of the worst atrocities of the war... the Dresden firebombing.
Billy believes that in 1944 he became unstuck in time. One reason that he is so incredibly nervous and awkward all the time is that he never knows which part of his life he is going to be performing next: an eye exam on a kid who's just lost his father in Vietnam, a speech to the Lions Club, his honeymoon with his wife, his stay in a prison hospital in the British compound of a POW camp in Germany, or one of the many other moments we're treated to in the book.
Billy's random zipping back and forth among different moments of his life gives us a plot-level reason to experience the novel in the "Tralfamadorian" way, as an apparently random series of moments without a definite beginning or end.
There is plenty of evidence throughout the novel that Billy is suffering from Post-Traumatic Stress Disorder (PTSD). We know Billy gets nightmares because when he falls asleep in the boxcar in Germany that's taking him to a POW camp, the other prisoners don't want to sleep next to him due to his whimpering and kicking. He startles easily: when he hears a siren going off in Chapter 3, Section 6, he jumps and worries that World War III is coming. And one of the most prominent symptoms of PTSD—the reliving of horrific past experiences—becomes literal in Billy's case as he travels in time.
But while Billy's numbness and exhaustion all fit into official symptoms of PTSD, his travels to Tralfamadore are something else entirely.
Billy's section of the novel starts out with the revelation that he believes he was abducted by a bunch of short green aliens from the planet Tralfamadore in 1967 to be part of a zoo exhibit on their planet. He goes on the radio and writes letters to the local newspaper explaining the Tralfamadorian concept of time. Yes—everyone thinks he's completely nuts.
The Tralfamadorians think that time does not go forward; instead, all points in time exist simultaneously. Nothing happens before or after anything else, so we cannot change anything and we never die. Sure, there are moments when we are dead or unborn, but those moments exist alongside our own living experiences. Billy feels that this should comfort us Earthlings who are afraid of death.
We get lots of indications that this is not meant to be a straight sci-fi kind of story and that Billy's account of his own experiences is unreliable. He only starts telling everyone about his experiences on Tralfamadore after he breaks his head open in a plane accident in 1968. Incidentally, this is kind of like the real-life experiences of the French author Céline, which the narrator mentions in the first chapter, who experiences "noises in his head" (1.20.2) after a head injury.
Billy's story of abduction appears to be strongly influenced by other events in the novel, so much so that it could be happening entirely inside his head rather than as part of the "real" world of the novel. For instance, his story of his life in a zoo on Tralfamadore seems to mimic the plot of a Kilgore Trout novel he reads in Chapter 9, Section 26. He watches a porn flick featuring the woman he claims to have been placed in captivity with, Montana Wildhack, in Chapter 9, Section 30. And the prayer engraved on Montana Wildhack's locket appears framed on the wall of Billy's real-life optometry office. (Check out our "Symbolism, Imagery, Allegory" section for more on the significance of Montana's locket.)
Yet even if Billy's ideas about Tralfamadore are imagined, they do point to something both he and the narrator desperately crave after the suffering they have experienced. Billy comments, and the narrator agrees, that he wants the following written on his tombstone: "Everything Was Beautiful, And Nothing Hurt" (5.51).
Only in the Tralfamadorian world can one cherry-pick the good moments from life to relive endlessly. The truth of the matter is that most human lives involve suffering. As a witness to terrible violence, Billy Pilgrim struggles more than most to find a way to explain how life can be so unfair and meaningless. Billy's trauma over the war is so severe that he has to leave Earth (either in his mind or for real) to find comfort after all the violence he has seen.
In the end, what matters more than the reality or unreality of Billy's life on Tralfamadore (after all, he's a fictional character, so the whole question is kind of moot) is his desperate longing for new ways to explain human suffering. The old models of religion and patriotism don't seem to solve anything for him. (For more on this subject, check out the "Fate and Free Will" theme.)
Even though most of the novel is focused on Billy, and he appears in nearly every section of every chapter, he is not what Vonnegut/the narrator would call a "character." Nothing happens to him of his own choosing: he gets drafted into war; he goes along with his marriage to the unattractive but well-meaning Valencia Pilgrim; he gets bullied by his daughter, Barbara. He is just not an assertive guy.
There is a reason for this lack of action, which the narrator very helpfully explains:
There are almost no characters in this story [...] and almost no dramatic confrontations, because most of the people in it are so sick and so much the listless playthings of enormous forces. (8.4.1)
Billy is too ill and weak to be an active character. Instead, he is forced to watch terrible things happen, things that are so awful that they send him out of time (either truly or in his own mind) to experience his own death many years before it happens (2.25.2).
The closest Billy comes to direct action is when he breaks out of the hospital to get on a radio talk show to spread the word about Tralfamadore. In other words, he becomes a "character" only after he goes completely crazy. And even then, it's to tell the world that he's been abducted against his will by aliens; he can't take credit for these ideas about life, the universe, and everything else.