Was ist der Unterschied im Englischen zwischen "killer, murderer and slayer"?

A killer is someone who kills. Usually this means other people. A soldier in the military is certainly a killer. But he is no murderer.

A murderer is someone who kills another person, and does so illegally - meaning he isn't a soldier in a war or an executioner or anyone else who would have legitimate cause to take a life. Every murderer is also a killer, but not the other way around.

A slayer is someone who kills with extensively. In fantasy literature, you certainly see the title "Slayer of Dragons". But you can also slay many other animals and even people. You would use this term for dramatic effect (and some newspapers have also done that, for precisely that reason). A slayer can be a murderer or a killer, if he slays people.