The object of arena game mode is to destroy all of the opponents. Each player gets one life per round, and the surviving team wins. If it proves difficult to kill the other team, after a set time, the one control point in the center of the map becomes unlocked and may be captured for victory.
Since there is no way of healing don't just shout for a healer, switch to a healer as they usually last the longest. If the other team is going rocketman, only then switch to rifleman or infiltrator. It is sometimes a good idea not to focus on killing everyone, but simply cap the point. Make sure you check to see if someone is idle instead of presuming its an infiltrator.
Generally infiltrators or healers last the longest, with runners being the least useful class. Rocketmen are very good, but can be killed by a good firebug or rifleman. Firebugs are good for killing those pesky infiltrators and Rocketmen. Overweights are good at defending the point on arena_aquaductv2.