Landa is probably a sociopath, but I'm not as certain he's a psychopath.
Sociopaths lack morality or a capacity for guilt or remorse and are able to do terrible things, such as violent murder. Psychopaths fit this description as well, but they have a necessity to express violent emotion and rage, often through murder, and can have a delusional perception of the world (though this isn't always true). A sociopath would probably never needlessly murder or betray someone if they couldn't gain personally from it, or if they weren't obliged to do so (perhaps if they were bored and the punishment would be light or non-existent), though in such situations they wouldn't hesitate. A psychopath, like most serial killers, uses violent acts as outlets of emotion and seems to enjoy violence for violence's sake.
What makes me hesitate is Landa's incredibly emotional murder of von Hammersmark, which he had no compelling reason to do. It may be, however, that she did violate some kind of code (not based on morality) that he did adhere to. Sociopaths and psychopaths can adhere to kinds of codes; Ted Bundy was a volunteer on some Republican campaigns and John Wayne Gacy was a staunch Democrat, not to mention history's many amoral members of various religious communities. If this is the case, he may have done it as a form of revenge on behalf of, a final act of loyalty to, or mental release from the party he formerly associated with, and simply because he knew there would be no retribution. And besides, Landa didn't actually personally kill anyone else in the film, though I think some psychopaths can enjoy murder by proxy (I'm thinking Manson).
This seems hypocritical considering his immediate betrayal, but I think Landa did have a sense of loyalty to the Nazis. I say this because at the opening of the movie he noted his pride in his "Jew Hunter" nickname and the nobility of the German people to the French citizen, whom he had no reason to lie to one way or another. At the end of the movie, he tells Aldo that he dislikes the same nickname, but appearing disconnected from the ideology of the Nazis better serves his purpose and his appearance of reliability in this case.
