Once you have a decent relationship with a lord, find him again and initiate a conversation. There should be a dialogue option along the lines of 'I wish to offer my services as a mercenary'. If your renown is high enough and your relationship with him is positive, he will accept your offer. You will then sign a mercenary contract, which typically lasts for a few weeks or months.
As a mercenary, you will receive a weekly wage from the faction, based on your renown and the size of your party. You are also free to attack enemies of your faction without penalty and can participate in sieges and field battles. This is an excellent way to earn money, gain experience, and acquire better equipment. Remember, you are not a vassal yet, so you don't own any land or swear fealty to the king.