Marvel and DC characters literally exist in one universe. It's not a one-time crossover event. They have shared histories, mutual friends, mutual enemies, and the like. For example, maybe Matt Murdock went to law school with Harvey Dent and Ted Kord and Tony Stark bump into one another at cocktail parties from time to time. 

So keep that in mind when filling in the "History" and "Supporting Cast" sections of your character sheet.