After reading How do I choose a faction you are ready to join the faction of your choice. You approach the leader of your chosen faction and say the word 'join'.

At this point the leader will assign you a task: Duke Akardius will ask you for a gem, Princess Kateriina will ask you to deliver a letter, and Jonas D'Accor will ask you for a weapon/armor item. Complete the task and you
will be added to the ranks of that faction and reap its benefits.

You can also join a faction by delivering a token to the faction leader.

Over time faction leader start to lose faith in you(unless you're a soldier.) In this case you will get a message telling you that your faction leader is no longer convinced of your loyalty. You will need to visit them again and say the word 'loyalty'. This will initiate another task that you must complete to continue to enjoy the faction's benefits.

You can also re-convince your liege of your loyalty by delivering a token to them.

Info by DogBreath