As someone who is entirely dissatisfied with his life and surroundings, Paul is predictably a misfit within his society who does a lot of things to demonstrate his dislike for his current situation.
The first evidence we get comes from Paul's teachers. The story opens up with a meeting regarding Paul's bad behavior. Arrogant and foppish, Paul shows up to the meeting wearing a red carnation and ill-fitting, flamboyant clothing. Far from showing class distinction, this unintended "costume" was yet another way for him to challenge them. It is clear that Paul hates his teachers just because they represent a system in which he feels ill at-ease.
Disorder and impertinence were among the offences named, yet each of his instructors felt that it was scarcely possible to put into words the real cause of the trouble, which lay in a sort of hysterically defiant manner of the boy's.
Throughout the story we also learn that Paul tells lies to his peers to add to the enigmatic persona that he wants to push forth onto others. He is happy to work at the Carnegie Hall as an usher. He is ecstatic to mingle with the actors and actresses, however, he shows off in typical teenager manner and embellishes the stories only to fall flat later.
When [...] his audience grew listless, he would bid all the boys good-bye, announcing that he was going to travel for awhile; going to Naples, to California, to Egypt. Then, next Monday, he would slip back, conscious and nervously smiling; his sister was ill, and he would have to defer his voyage until spring.
Paul shows contempt for his neighbors and family members. He just cannot stand commonalities such as Sunday brunches, or family gatherings. He is more about the artificiality and aesthetics of daily living. To him, normal life in Cordelia street is tedious and he will let everyone know with his stooping attitude that he is not happy there.
Once out of Cordelia, and into the Carnegie, Paul "lived again". He was good friends with Charley Edwards, an actor. He behaves completely different with the acting troop because he feels that it is all part of a big stage and that he is in it. He did not want to be an actor, just a part of "a scene". The scene, however, would not be of a play either. Paul's view of life is that it should always be pomp and revelry; that he is, or should be, a part of that.
Finally, Paul takes his own life after escaping to New York City and living for two days at the Waldorf Astoria under the guise of being a rich heir. The idea of going back to the common world disgusted him beyond belief. Moreover, he did not want to deal with his father, whom he had already disappointed so many times before.
This shows a side of Paul that may have still a degree of consciousness and humanity. He may not be this enigmatic entity that he wishes to portray, but a lost teenager who has no clue what he wants from life. We will never know, however, but the way that Cather describes his death shows that he may be able to find himself
....into the immense design of things.