There are several traits that could be picked up when your characters are already placed on the throne. Namingly, they are "diligent", "gregarious" and "just".
The "diligent" trait could be picked up by going on the hunt, an event often fires where you can pursue prey and automatically receive the diligent trait (not available to Muslims) Although, to obtain this diligent trait, there's a 5% chance of getting wounded.
The "gregarious" and "just" trait could be picked up by organising a feast (not available to Muslims either). When there's a scenario when your vassal slit the throat of your servant, you could imprison the vassal, in exchange, your character will gain the "just" trait. As for the "gregarious" trait, you can only gain it when the scenario of joining your vassals in the dance appears.
Gaining these traits are advantageous as they gave you +10 vassal opinion for each trait. Therefore, you might want to go to hunt and hold a feast every now and then.