To become a mercenary, you first need to gain some renown and build a positive relationship with a lord from your desired faction. You can do this by completing quests for lords (found in castles and towns), participating in battles where they are present, or by defeating enemies of their faction. Look for lords with a 'friendly' or 'good-natured' personality, as they are generally easier to impress.
Quests from lords often involve hunting down bandits, delivering messages, or collecting taxes. Successfully completing these will increase your renown and your relationship with that specific lord. Aim for at least +5 relationship with a lord before attempting to offer your services. You can check your relationship with any lord in the 'Reports' menu under 'Characters'.