How do I start my own faction in Mount&Blade?

To create your own kingdom, you must conquer a castle, any castle. First, you have to begin a war with an existing kingdom. For this, you could try attacking caravan or pillaging villages until you are under 0 relationship with the kingdom. Then, you can start attacking castle and cities.

IIRC, you can also , if you are a vassal to a kingdom, start a rebellion. You have to ask to be given a land, and, if refused, challenge the king and start a rebellion.. If you are wife to a lord of a kingdom, you can also ask him to join you in rebellion, thus starting with some land.

Once you have your castle, you can start giving titles to your companions. When you do this, he will quit your party and patrol around the village, castle or town you gave him, until you give him another order. You also have to put garrisons in your castle and town, otherwise you'll lose them very quickly.