Someone whose solution to every problem is, "kill it", regardless of the appropriateness of that solution. Kill an orc that tried to ambush you in a dungeon? That's normal, and not murderhobo. Kill the guild leader because he was annoying you? That's murderhobo.
The 'hobo' part implies lack of a home, but it really should be taken as more abstract than that. It's not that an adventurer doesn't have "a home". It's that he doesn't have a home here. The adventurer doesn't have anything at stake 'here' (even if he has a home somewhere else), and thus feels that there are no consequences for his behavior. It's the ultimate tourist syndrome. "What happens in Vegas stays in Vegas."
The adventurer doesn't have a mom to guilt-trip him, or a dad to be ashamed of him, or children to set an example for. At some point, he reaches that consequence-free nirvana, and becomes an internet troll with a sword. That's the point he becomes a murderhobo.
