The psychopathic character endlessly fascinates the darker side of human nature. While psychopaths in real-life are best to be avoided for a whole host of reasons (more on how to spot a psychopath here), it is the fictional psychopaths, such as those that stand out in movies, that many of us can't get enough of.
Psychopathy is understood as a psychological syndrome, or a series of traits, that exist in a small fraction of the population. The stand-out feature of this collection of traits is a lack of empathy, or understanding of basic human emotions. After that, psychologists would identify the chaotic nature of a psychopathic individual (i.e. instinctive actions lacking in any planning, a lack of any concrete life "goal"), and the cunning and amoral use of manipulation to get what they want, without regard to the consequences for others. In essence, a psychopath has no moral (i.e. empathy) compass, and he will whatever is easiest and convenient to satisfy his needs at that time. There are many traits, but those mentioned are likely to be the most prominent.
An agent of chaos
A feature-length British documentary by Channel Four (called "Psychopath Night") used some experts in the field of crime and psychiatry to look in some depth at clarifying what a "psychopath" exactly is. One of the experts was Kevin Dutton, who described his "mixing-deck" approach to understanding how "psychopathy" may manifest itself throughout a section of human society.
At the same time, the documentary used the experts to analyse what they thought were the most convincing portrayals of the psychopathic character in film. Heath Ledger's playing of "The Joker" was considered best, as explained here (watch from 8.37).
As mentioned by the former FBI expert in the documentary, in the bank robbery scene at the start of the movie, The Joker shows callous duplicity and a complete lack of regard for his cohorts - having them kill each other one-by-one to apparently get more of the money, until there is only The Joker himself remaining. In the same scene, he displays grandiosity in the dramatic and terrifying nature of the bank heist, clearly wanting to make a statement of intent (and which is later demonstrated when he later crashes Bruce Wayne's party to get Harvey Dent). Lastly, in the conversation with the bank manager, he makes it clear how nihilistic is his world-view; he has no moral code, only a belief that what doesn't kill you "makes you...stranger".
Throughout the film, The Joker does whatever is necessary to achieve what he wants: though often this appears to be to cause chaos simply for the sheer hell of it. As he explain to Batman when interrogated, he is not a monster but simply "ahead of the curve". This is a purely amoral, psychopathic perspective of the world; an utterly cynical view of human nature, where he believes that people will turn on each other when the chips are down.
While he targets the main individuals working against the mafia - the judge of the mafia trial, the police commissioner, Harvey Dent, and the mayor - it is also clear that his wider purpose is to cause as much moral chaos in the city as possible. One the last scenes is where he creates his own kind of amoral, real-life social experiment: the ferry scene, where two ferries - one carrying civilians, the other with convicts and prison guards - are told to blow up the other before a deadline, or they would both be blown up.
He blows up a hospital, it seems simply for the sake of it, taking its patients as hostages, and after causing the explosion that disfigured Harvey Dent, in the final scene with Batman, The Joker declares it was his intention to see Harvey Dent, the city's "hero", come down to earth, in order to morally corrupt the city. In the later film, "The Dark Knight Rises", this prophecy finally comes true, when Bane reads out Gordon's speech (more on the psychological comparison between The Joker and Bane here). In the same way, he also brings about the collapse of the public's moral view of Batman himself, when Batman takes the blame for killing Dent and the corrupt police officers. This demonstrates that by the end of the film, although The Joker had "failed" in his main task (the story implies he is after complete criminal control of the city), he had succeeded in destroying Batman as a moral force for good in the public's eyes, and achieved his own kind of "moral" victory even though he would be behind bars.
As a psychological portrait, The Joker fits many of the traits consistent with a psychopath. It is in the hospital scene with Harvey Dent where the Joker makes his remark that he is "an agent of chaos", asking "do I look like the kind of guy with a plan?" This comment is further clarification of the chaotic and impulsive nature of his personality.
Master of Manipulation
As the film progresses, each scene with The Joker clarifies further that his character features strongly manipulative and cunning elements. He apparently seems to offer his services to the mafia - to kill Batman - while at the same time, destroying them as an organisation, one by one, culminating in burning half of their money.
He conspires events to have himself arrested in order to grab the mafia's accountant, Lau, by blowing up the main police station. While under interrogation with Batman in the station, he also tries to manipulate Batman towards his side, by insinuating that Batman is being used by the police, and will be blamed as a villain later (which is indeed what happens at the end of the film). He also succeeds in manipulating Harvey Dent - after causing the explosion that facially disfigured him - to go after the corrupt police who were working with the mafia boss Maroni, diverting the blame away from himself.
The Joker's real purpose, apart from causing as much mayhem as possible, is to ultimately take control of the city, as its chaotic criminal head. From gaining the attention of the mafia by robbing one of their banks, then offering them "his services" to kill Batman, he also succeeds in killing off his main mafia rivals, one by one. While he is supposedly working as an agent of the mafia in targeting the city's great and good, he also clearly has a plan of his own. First by killing Gambol who has already threatened to eliminate him, then, after burning half of the mafia's money, killing The Chechen. Lastly, as mentioned before, he manipulates Harvey Dent into killing the last remaining big mafia boss, Maroni.
In this sense, he Joker easily fulfills the portrait of psychopath as Machiavellian schemer, using the mafia's influence to target the "moral core" of Gotham City, while equally scheming to manipulate Harvey Dent and Batman to for his own ends. The Joker is a criminal mastermind lurking behind a dark facade of amoral nihilism.
In this respect, it is clear that if The Joker were a real human being, he would be a truly terrifying person to behold. While this character is an extreme (and fictitious) example, there exist real-life examples, too. Some of them are in prison (Ian Brady, for example), and others had their notoriety only fully revealed after their death (e.g. Jimmy Savile). However, you do not need to look very far.
In some respects, the personality of The Joker bears some similarity to that of Stalin: certainly, Stalin is a prime example of how a psychopath does what is necessary to get to the top, and what happens when a psychopath rules the largest country in the world.
When psychopaths are in charge, the result is chaos, and terrifying for everyone else: how do you think the financial crisis was possible?