Personal Contention Points are akin to lottery tickets for invitations to a port battle. Just because you have the most personal points, doesn't mean you will get into the fight. Conversely, just because you only have 1 point, doesn't mean you will be overlooked. However, the more points you have, the more chances you have to take part in the port battle.
Personal contention points on foreign ports are earned by creating unrest there through sinking ships of the nation owning the port or by blockading it. Points are earned on friendly ports by removing unrest caused by foreign nations, which can be done by sinking the ships of the nation holding unrest on the port or by patrolling the port.
Points can also be earned by providing unrest supplies to the Garrison Commander (on friendly ports) or the Rebel Agent (on foreign ports).
Unused personal contention points can be turned into General Contention Points for future use at other port battles.