What part of Holden's personality leads to his mental breakdown?
That is a complex question. It is probably all the parts of Holden's personality, plus some aspects of the world around him, that work together to orchestrate his breakdown. I will just focus on two parts of his personality, though.
Holden is sensitive and idealistic. Although he talks as if he's tough and cynical, he's actually very perceptive about injustice in the world and the vulnerability of certain people to it (including children and guys with acne). Holden is so perceptive and sensitive that these things really bother him. He would like to be able to rescue vulnerable people from injustice and tragedy, as if he were "catching" children who are about to fall off a cliff. Most of the time, of course, this isn't possible. Because of his sensitivity and idealism, Holden isn't just able to write these things off. Instead, he ruminates on them, digging himself deeper and deeper into depression.
Holden is restless. He has a low tolerance for mental pain. Throughout the story, whenever Holden is bothered by something, he takes some kind of action to get away from the mental pain or distract himself from it. He might spontaneously decide to leave the place he is in, he might call up a friend or acquaintance to chat, or he might visit a bar to drink, and flirt with women. These are all attempts to get away from the mental pain. Most people can identify with this. Unfortunately, in Holden's case, his self-soothing attempts make things worse. They lead him to become drunk, hungry, sleep-deprived, broke, and subject to a string of frustrating and fruitless conversations. His running behavior makes him feel worse, so he runs on to something else, which makes him feel even worse. It's a vicious cycle. The cycle continues until he meets up with his sister Phoebe.