The name says what you have to do. You have to complete 15 different characters mission modes storyline missions. You don't have to do the other ones unless you want to. There are usually 5 story missions for each charcter but I think that some characters might have less than 5 story missions. If you can't start story missions or there are no more available after you complete some than you should check the terminal because there will probably be a message that will unlock the next mission or missions. To do this quick and easy than you should do all the story missions on easy and you should use characters from official mode (if you completed official mode) because they don't start at level 1. If you don't have 15 characters in mission which I think you do (I don't remember how many characters you start with) then you should complete character's official mode because they unlock characters in mission mode.