The psychoanalytic approach can be applied to A Clockwork Orange in two ways. One way is to try and analyze the book itself, as well as its protagonist Alex. The other is to see how it is used within the book—in the form of the Ludovico Technique.
Firstly, the novel itself has been a very common subject of psychoanalysis and a lot of the critique it received revolved around this. Alex is a deeply disturbed individual, who can safely be called a sociopath. He is incapable of feeling any real empathy for his victims or taking responsibility for the suffering he causes. To that end, what's wrong with him is quite clear.
Other characters in the novel are a different case, however. Having a protagonist like Alex works kind of like a cover for the rest—compared to him, the others seem less bad. On closer inspection, though, there are very few genuinely positive characters in the book. Most of them submit to the same impulses, but to a lesser degree. In Part 3, Alex goes through several brutal beatings, all from people he's hurt earlier. Deserved? Yes. Right? Hard to say. Anger is a powerful emotion and Alex's crimes were terrible. From a psychoanalytical standpoint, Alex represents an almost pure Id—he acts to satisfy his instincts, as that's the only thing he cares about. If we are to agree that the rest of the characters are less "broken" in that sense—that is, they have all three parts of the psyche (id, ego, superego) working properly—it leaves the reader to assume they choose to let the anger rule them.
That is most obvious in the case of the writer whose house Alex burgles in Part 1. He is beaten and his wife is raped by the droogs. In Part 3, we find out she's died from her injuries. Now, when the writer finds a severely beaten Alex on his doorstep, he doesn't recognize him. Before the personal level comes to play, the writer is helpful. He dislikes the government and wants to use Alex as an example of brutality (since he was subjected to the Ludovico Technique). Then Alex reveals himself and everything changes. In psychoanalytical terms, the Id kicks in with full force and drives the Superego away. All other plans go out the door and only revenge remains. He ends up locking Alex in a room and blasts classical music to him—the thing that triggers the nausea and pain.
If the reader didn't know who Alex was either, we'd think it was cruel. But we do know who he is. Everything comes down to what we know and how we choose to act upon that knowledge, just like psychoanalysis suggests.
Secondly, psychoanalysis is used within the book itself. The Ludovico Technique is a severe form of aversion therapy, which is intended to physically stop Alex from committing crimes. As he watches violent imagery, he is injected with drugs to make him ill. It works. Alex becomes, for a while at least, incapable of harming anyone.
The way the aversion therapy is used here makes it a backwards version of free association, an important tool for psychoanalysis. Normally, free association is used to bring things from the subconscious to the conscious mind. The process also works naturally—smells, for example, are powerful triggers. When we smell something very particular, we're often reminded of a person, place, or an event. In psychoanalytical therapy, the method is used to find those sort of links and use the knowledge to our advantage.
With aversion therapy, the process works the other way around—the triggers are created. As Alex tries to do something violent, his body reacts and he becomes nauseous. The therapy forces something into the subconscious, instead of bringing it out. In real world, there are a lot of ethical issues with this, of course. Even in the book, the question of losing free will is raised. That is why real patients volunteer for it.