I killed a particularly difficult purple monster, almost dying and chugging all of my potions. After I killed it, it said 'bad fame'. What does this mean?
When you kill purple monsters (aka champions) you get fame. Fame is like a second experience bar that when full, grants another skill point to assign. The levels have names like unknown, remarkable, etc.
I'm guessing that you are just starting the game and currently your fame is low and is labeled bad. This isn't a negative thing, just currently the name of your fame level. If you gained more fame and got another skill point it would change to another name.
Also see: What does the fame stat do?