This paper explores character-driven story continuation, in which the story emerges through characters' first- and second-person narration as well as dialogue---requiring models to select language that is consistent with a character's persona and their relationships with other characters while following and advancing the story. We hypothesize that a multi-task model that trains on character dialogue plus character relationship information improves transformer-based story continuation. To this end, we extend the Critical Role Dungeons and Dragons Dataset (Rameshkumar and Bailey, 2020)---consisting of dialogue transcripts of people collaboratively telling a story while playing the role-playing game Dungeons and Dragons---with automatically extracted relationships between each pair of interacting characters as well as their personas. A series of ablations lend evidence to our hypothesis, showing that our multi-task model using character relationships improves story continuation accuracy over strong baselines.