Head to the Hero Quest counter in the lobby and select Solo quests if you are given the option. If not, it might be because you don't have the Dissonance of the nexus dlc which adds party hero quests and will automatically go to the solo quest list.
There's a total of 4 different Maps that unlock as you progress through the main story. To get the achievement, finish each one once with any character of your liking. You do not have to finish each with every character.