A Multiplayer Campaign essentially is an online "match". From the main menu, select Gladiator and choose Smart Match or Custom Match. Custom Match will allow you to pick specifically what Arena Playlist you want to play, whereas Smart Match is more like "Quick Match" from other games.
Each Arena is a playlist with usually 2, sometimes 3, different "Rounds" involved. When a round is over, you will be greeted with a stats page based on that round. You will only have one option on this screen, a "Continue" button at the bottom. After a brief pause, you will continue on to the next round which is noted by the arena environment changing in a cut-scene shortly before gaining control. When the actual match is over, you will go to an "overall stats" screen after the end-of-round stats.
NOTE - I read over on Xbox 360 Achievements that it can be obtained in solo or co-op play, win or lose. Credit to X360A user Metabolic x360a