What do you call a person who hates everyone but himself?

I believe the correct word would be misanthrope or misanthropist.

As @theUg suggests, I am elaborating my explanation. Following are the reasons why I believe misanthrope should be the correct word here:

• Its definition (source - Wikipedia)-

Misanthropy is the general hatred, mistrust or disdain of the human species or human nature. A misanthrope, or misanthropist is someone who holds such view or feeling. The word's origin is from Greek words μῖσος (misos, "hatred") and ἄνθρωπος (anthrōpos, "man, human").

• Scrooge as @JAM suggests could also be considered correct but not 100%, because the dictionary defines it as a person who is miserly. Also misanthrope looks much more formal. Same is the case with words suggested by others as they are not 100% correct for this situation.

Note: this echoes several other answers and comments, but I had felt most of them lacked detail, and as I became interested in this topic I decided to elaborate.

Short answer

A misanthrope would be the most fitting term, but with large shadow of philosophy hanging over, it may require deeper understanding, although in common usage and less demanding context, the somewhat pedantic details may be overlooked.

Long Answer

First ingredient in our recipe would be the “hatred of mankind”. Dictionaries differ in their interpretation of misanthropy, from mere “dislike” in OED’s definition, to “mistrust”, to the full-on “hatred” in M-W’s. But the etymology is unambiguously clear:

from Greek words μῖσος (misos, “hatred”) and ἄνθρωπος (anthrōpos, “man, human”)

With that taken care of, the next step is exclusion of self from universal loathing. That is where things become complicated, as misanthropy can come in many flavours, and, on the other hand, some ideas that may look like misanthropy, are not (that is why M-W’s synonymizing of misanthrope with the words like cynic, or pessimist is less than responsible).

One early definition of misanthropy can be found in Plato’s dialogue Phaedo:

For misanthropy is clothed out of trusting someone excessively without skill, and believing a person to be completely true and sound and trustworthy, then a little later finding him bad and untrustworthy and again with another; and when someone experiences this many times and especially by those who one believes are nearest and dearest, so often taking offence he ends up hating everyone and believes absolutely no one is sound at all.

In essence, it is about someone placing unrealistic expectations on others, and then crushing disillusionment when those fail. It does not imply that the doubts of the worth of humankind extend on the subject, but it may force one to separate from society. However, that would be a key point, as according to Aristotle’s Politics, a solitary man is not a man at all (from 1598 translation titled Aristotles Politiques, or Discourses of Government):

…he that cannot abide to live in company, or through sufficiency hath need of nothing, is not esteemed a part or member of a city, but is either a beast or god.

Thus, Aristotelian qualification would allow a reclusive misanthrope to avoid a pitfall of intellectual dishonesty (“I hate all of the humanity, except myself, though I am part of that very humanity”), or becoming a pitiful wreck as Alceste, the character from Molière’s satire Le Misanthrope, had. And, on the subject of gods, one can be auto-theistic misanthrope. Nietzschean idea of Übermensch and other forms of radical individualism were also criticized for that sort of disregard for humanity.

On the other hand, even though Schopenhauer proclaimed “human existence must be a kind of error”, misanthropy does not necessarily mean malicious hatred of others. We could all be failed beings, doomed to suffer together.

Finally, why I think the sorts of narcissistic misanthrope would not work. Basically, it is an oxymoron. Misanthrope despises humankind, and dislikes the idea of society, but narcissists, “unethical” egoists, megalomaniacs and other radically-selfish types, either strive for validation from society, or its subjugation or exploitation.

Like many other answers here, this one is not perfect, but how about

sociopath

This was my initial response, and a simple Google search reveals that sociopathy is thought of as a severe form of antisocial personality disorder, so it seems like Jim and I are on the same track.

From Merriam-Webster online:

so·cio·path
noun \ˈsō-sē-ə-ˌpath, ˈsō-sh(ē-)ə-\ : a sociopathic individual

so·cio·path·ic
adjective \ˌsō-sē-ə-ˈpa-thik, ˌsō-sh(ē-)ə-\ : of, relating to, or characterized by asocial or antisocial behavior or exhibiting antisocial personality disorder

an·ti·so·cial
adjective \ˌan-tē-ˈsō-shəl, ˌan-ˌtī-\ : averse to the society of others

Though I have no evidence besides various episodes of televised legal dramas, my sense of sociopaths is that they feel somewhat neutral about themselves. The aspect of non-self-loathing you want to communicate with this word seems to me incorporated in sociopath. There is a term, narcissistic sociopath, which seems to explicitly communicate self-aggrandizement (which you didn't state as a requirement for the word).

I'm not sure if my assessment of a sociopath is right, and it may be too strong a word for what you intended, and also come with some connotations about destructive behavior that you do not intend, but that's all I got. As stated, it seems English has no perfect word to answer your question. Misanthrope, suggested above, is the next closest thing I can think of to the suggestions I have given.

For reference, here's a very interesting breakdown of the related psychological disorders

In one word it would be antisocial.

'Narcissist' (defined as 'someone who loves oneself excessively') seems to be the word that fits your description.
'Hating everybody else' would be the result of a state of extreme narcissism.