How did Holden start hating society?

Sorry, your brother just died. You can't come to the funeral though, because your hands are all injured. In a few years, you'll be sent to one of the most expensive boarding schools in the country. Make sure you study hard so you can get into Yale and make good friends there. Good luck!

This is probably what happened to Holden after Allie's death. He was prevented from attending his beloved brother's funeral, and with no support system to help him cope with his grief, he was thrown into a competitive environment of boys who were snobby, stole his things, manipulated him (cough cough Stradlater), and never really cared about how he felt (Ackley...). From this perspective, we can better understand how Holden came to form his complaints about society, which I broke down into three main categories: 

• Social standard of success  
• Contempt for other people's misbehavior/stupidity 
• Loss of genuineness in life and personal connections 

Social standards of success 

Holden's rich family and the privileged kids at his school present a model of success in society, composed of material wealth (ex. cars, money) and prestige (ex. attending a prestigious college, becoming a lawyer). Holden is told, and perhaps would have formerly believed, that he must study hard and follow the conventional path to become "successful", which supposedly is the purpose of his life. However, the death of Allie changed everything. The loss of one of the most valuable people in his life caused Holden to realize the volatility of external standards of success in the face of death and hence convinced him of the meaninglessness of putting in the effort to attain material wealth/prestige. 

As a result, he started to stop "applying" at his school and felt indifferent about losing/giving away his material possessions (ex. losing his gloves, giving his money to the nuns). At the same time, as he yearned to move away from his elite life, he faced (and is still facing) ambiguity about his future. One manifestation of the anxiety around his future can be seen in his repeated inquiry about where the ducks in the lake would go in the winter--a metaphor for what would happen to him at the end of high school when he would have to face the consequences of failing at school (during the "winter" of his life). 

Contempt for other people's misbehavior/stupidity

Holden sees Allie as one of the smartest and nicest people he'd ever known. According to Holden, Allie is very mature and clever (ex. he didn't get sore when Holden didn't let him go shooting BB guns with him). Now, that Allie is gone. In the eyes of Holden, everyone else who is alive (except a select few, like Phoebe) are much worse human beings than Allie. The boys at his school and the Ivy leaguers he sees at night clubs are petty people who get pissed if they lose in games, mistreat girls, and are busy showing off their knowledge/skills (ex. criticizing movies). Holden feels that everyone is so engaged in "boring" things like movies and that there is no one smart enough to entertain him with an intellectual conversation (or share his enthusiasm about the kettle drummer at the Christmas concert like he did with Allie). 

Loss of genuineness in life and personal connections

Loss of genuine personal connections 

Allie's death mentally ostracized Holden from everyone else at school. While everyone was talking about sex and liquor, Holden had internally been suffering from his grief of Allie's death and no one actually gave him useful guidance on how to cope with it. Therefore, he had to figure out his own way of dealing with grief: alcohol and smoking (which his mom also does a lot). Throughout the novel, we see Holden continually using these two things to give him immediate relief from any sense of depression or loneliness creeping on him (ex. after his argument with Sally, he intentionally gets as drunk as he possibly can to alleviate t depression). As he became increasingly reliant on alcohol and smoking, he also increasingly avoided directly confronting his emotional problem. Aside from talking to Sally, he rarely opens up to anyone else about his internal problems to receive help. Also, he hides much of his internal thoughts and feelings towards the people he interacts with, often displaying a more polite, neutral facade in front of his "friends" behind all the things he criticizes them for. This led Holden to feel that others are being phony (when actually he was the one being ingenuine to others). 

(To add on, Holden ultimately feels lonely at every school/place he goes to as a result of this emotional barrier he builds and ends up literally begging people to hold a conversation with him, even at the expense of talking about something he's not very interested in, like sex with Luce. He also never resolved his grief toward Allie's death, which we can see from his habit of talking to Allie whenever he's very depressed.)

Loss of a genuine life

As discussed above, Holden feels that the external markers of success like money, prestige, and social belonging are meaningless. As a result, he criticizes other people who are "wasting" their lives chasing these things (like D.B. being a "prostitute" wasting his talent in Hollywood). He dreads the picture of himself living the normal life of a "successful" salaryman, going up and down his apartment on an elevator and sending phony cards to everyone at a hotel. He thinks such a life would be too boring and phony for him. 

(In addition, the death of his brother at such a young age would have created a sense of urgency to chase his genuine source of happiness and passion in the "short" amount of time he has left in his life.)