Once you have sufficient renown and a good relationship with the king (or a lord who can recommend you), find the king in his castle or town. Initiate a conversation and express your desire to swear fealty. If he accepts, you will become a vassal of his kingdom. This is a permanent decision for that playthrough, so choose your faction wisely.
Upon becoming a vassal, you will likely be granted a village as your first fief. This village will generate income for you, but you will also be responsible for its defense and prosperity. You can appoint a companion as your minister to manage your fiefs and other administrative tasks. As a vassal, you can also recruit troops from your own villages and castles, and participate in the kingdom's political system.