What are some ways to approach a character analysis for the novel The Catcher in the Rye?

One way you may want to approach your character analysis of Holden is to actually conduct a psychoanalysis of him. (Here's a of this theory.)

The reason why this type of character analysis might work is that it's suggested on the opening page ("I got pretty run-down and had to come out here and take it easy"), but confirmed at the novel's end, that Holden himself is undergoing therapy, most likely psychoanalysis.

What you would probably want to do is look at the Freudian concepts, particularly id, ego and superego. Simply put, the id is the unconscious part of the human psyche that responds to instincts and human desires (sex, food). The ego is the mediator between the id and the real world (from SimplyPsychology):

The ego operates according to the reality principle, working out realistic ways of satisfying the id’s demands, often compromising or postponing satisfaction to avoid negative consequences of society. The ego considers social realities and norms, etiquette and rules in deciding how to behave.

And the superego functions to create the feelings of right and wrong and moral and immoral in a person. Guilt comes from the superego.

I would probably focus on how Holden's superego is his dominant psychological force. He thinks of things in terms of what's right and wrong. For example, according to Holden, Stradlater shouldn't have sex with Jane Gallagher because he doesn't know about how she keeps all her kings in the back row. Throughout the novel, Holden's superego battles with his id. While Holden refuses to have sex, he is driven by that human desire. This is why he hires a prostitute and calls Sally Hayes. In each of these instances, Holden's superego, unconsciously, convinces him that it's wrong. He wants to talk to the prostitute, not have sex with her. He asks Sally to run off with him because it's pretty clear from them making out in the cab and Holden telling her he "loved her and all" that his relationship with her, especially if he went over to "trim the tree Christmas Eve," could lead to his first real sexual encounter. So, it's possible that when he asks her to run off with him, he's purposely sabotaging the relationship.

Obviously, this is a different type of character analysis, but it can be effective.