As a king, how can I improve the relationship between my faction and other factions without going to war with other factions?
If you are the king of a faction, there are multiple ways to improve relationships.
Bribe them: The first is the easiest but costly, talk to your minister and choose to send a faction a gift. Then just let time pass and your relationship with the target faction will be bumped a few points.
Rescue them: Often, faction lords tend to get captured. If you go find the castle they are imprisoned at, break them out of jail (walk around town/castle, find the jailer, pick a fight with them, then go into the jail and break the lord out, finally kill all guards to escape). This will improve your relationship with the lord and the faction they belong to.
Join their fights: If you see a lord in a fight, doesn't matter against what (other factions, bandits, etc), just join the fight on their side for a boost (assuming you win). This is a sneaky good way of reducing the army size of a kingdom you are planning to fight as this doesn't damage relationships with anyone since you are simply coming to the aid of a 'friend'.
Make a treaty with them: This is the least probable but the easiest way to bump some relationships. Send a companion with a high persuasion to try and make a truce or treaty.