From the main menu , choose "STORY: Soul Chronicle" At the top of the next menu is the main story, a horizontal line with 20 dots on it. Each dot is a battle you must complete to further the main story.
Most of the storyline fights are simple, but might force you to use a character you're unfamiliar with. Try practicing with a character's individual storyline, below the main story. These aren't required for the achievement, but serve as an introduction to the character and their moveset.