It is pretty much standard practice for fiction writers as well as dramatists to introduce their characters doing something that is characteristic. (For example, Hamlet is introduced brooding and dressed all in black mourning clothes.)
Holden is introduced standing all alone on top of a hill in the cold, looking down at the latest school from which he has been expelled. He is a lonely outsider--and his condition has been made worse by the fact that he has been "ostracized" by the fencing team he managed--or mismanaged. They had gone into New York for a match and he had lost all their equipment on the subway. This too characterizes Holden. He is absent-minded because he gets lost in his own thoughts and fantasies--and this is undoubtedly because he is extremely intelligent, although he thinks of himself as dumb because he flunks out of schools. His intelligence is shown especially in his being credited with writing the whole text of The Catcher in the Rye.
He explains--or tries to explain--why he is really standing up there on Wheaton Hill:
What I was really hanging around for, I was trying to feel some kind of a good-by. I mean I've left schools and places I didn't even know I was leaving them. I hate that. I don't care if it's a sad good-by or a bad good-by, but when I leave a place I like to know I'm leaving it. If you don't you feel even worse.
These words show that Holden is a sensitive boy with strong feelings, as will be demonstrated throughout the novel. He attaches importance to feelings. He is really feeling terrible. He feels disgraced, not only because he is flunking out of Pencey Prep, but because this is the third or fourth school that has expelled him for poor academic performance. Not only that, but he has disgraced himself with the fencing team. He has a bad reputation in the school with both the students and faculty. He is a nonconformist in a setting which emphasizes conformity. He might be called a scapegoat. The author, J. D. Salinger, told people that Holden Caulfield was a portrait of himself as an adolescent, a fact which should be obvious enough without his acknowledging it. Salinger too was an introvert, a genius, and a nonconformist.
Holden really seems to be trying to feel that Pencey Prep was his school--that he did indeed belong there, even though he is effectively no longer a member. He is in a funny position: he is still living in the dormitory and eating his meals there, but he has become a non-person. He is practically invisible--like a ghost haunting the place. He only has a few more days to stay. It doesn't matter whether he attends classes or not. Nobody wants him. And if he doesn't belong there, where does he belong? He is going to have to face his parents (who probably don't want him either). This is a coming ordeal he doesn't even want to think about, and therefore he doesn't mention it until Mr. Spencer brings it up in Chapter 2.
"And how do you think they'll take the news?"
"Well . . . they'll be pretty irritated about it," I said. "This is about the fourth school I've gone to."
Holden's decision to go into Manhattan is obviously rash. He is only sixteen. He leaves Pencey because he feels totally out of place and unwanted. He finally gets to say his good-by at the end of Chapter 7.
I was sort of crying. I don't know why. I put my red hunting hat on, and turned the peak around to the back, the way I liked it, and then I yelled at the top of my goddamn voice, "Sleep tight, ya morons!"