To complete a round, you must only reach the concluding score screen. It doesn't matter whether you win or lose. Simply start and finish 15 rounds in Gladiator Mode and you will complete the challenge.
Additionally, even thought the description says "in Multplayer," this can be done in Solo mode. Just start up any of the arenas solo and let yourself die 15 times and the challenge is yours. I killed a few enemies each time for good measure, but I'm sure this isn't necessary for the challenge.
One last thing: I'm fairly certain that "15 Rounds" means exactly that. Each part of an arena mode playlist should count as a round toward this. I am NOT entirely sure this is the case. It would be greatly appreciated if someone could confirm or deny.