人殺し: Murderer. Killer. The easiest 和語 (native Japanese) term. Often chosen when yelling at someone (人殺し! "Murderer!"). Also refers to the act of murdering (killing; murder case).
殺人犯: Murderer (as a criminal, as 犯 suggests).
殺害者: Murderer. Killer.
殺人犯 and 殺害者 are sino-Japanese words. They sound more technical, and are preferred in formal settings or detective dramas. 殺害者 is relatively uncommon.
殺し屋: Professional hitman who kills anyone for money.
刺客: Assassin. Someone sent to somewhere to kill a certain important person. Often used figuratively to refer to someone who "kills someone socially/politically/economically," for example, a strong opposing candidate in election.
暗殺者: Assassin. Usually used with very important person like a diplomat or president.
兇手/兇漢: I haven't seen these words, so I think you can forget them :)
Another word worth memorizing would be 殺人鬼, which refers to a devilish (serial) killer like Jack the Ripper.