I have a story in mind in this setting, that only uses the concept of the DBZ world, like with Ki, Saiyans and so on. My character would be a half-Saiyan, living with his mother, whom was raped eighteen years ago during the first Saiyan invasion. Now... a second one is heading for Earth, along with many, many other threats, like crazed world-conquerors, tyrants wanting immortality and general run-of-the-mill crooks that somehow learned secret martial art-techniques. Your character would be a girl going to the same school as we, who also has some marital arts training. Will she work out the mystery of the Legendary Z-fighters, the Seven Protectors chosen by Kami to each guar the location of a Dragonball. Will my character become one, despite the distrust of the others for his half-alien heritage? Will your character find out whom the powerful fighter is that helps out, yet keeps his identity a secret? Will she strive to be a Z-fighter herslef? Or simply want to win the World Tournament?
Suggestions, variations, or new stories with the DragonballZ setting are all possible.
Though, if we head into canon world, I like to play Gohan.