Mount & Blade: Warband

The Basics:
For each faction there is a claimant who says that the throne rightfully belongs to them. Travelers in the tavern can give you the location of these claimants and once you have enough renown, you can vow to help them reclaim the throne.

Once you take up their cause, they will join your party as a companion and you will be the leader of the rebellion against their desired faction. From there you will need to slowly capture the other faction's castles and towns and win over lords to fight for your cause. Capture all of the other faction's territory and their ruler to force the faction to disband. This can take a while as lords still loyal to the faction will continue to roam around and siege your castles for some time, even after the faction has no territory. Capturing them after battles will speed this part up though setting them free will increase your relation with them and hopefully make them more likely to join you.

Once they finally disband, the claimant will leave your party and become the new ruler of the newly reformed faction. Your rebellion faction (red color) will disappear and you will become a vassal of the faction you've just conquered. At some point later, you will likely be made the marshal for the faction too which would earn you the Autonomous Collective achievement.

Note:
Thanks to Sky Maniac's Empress achievement guide - You need to capture a faction's ruler before capturing all of its territory to avoid a bug where the ruler disappears from the map. If this happens, the faction will never disband and you won't be able to get the achievement with this particular claimant.

Advice:
I had a lot of trouble in the beginning of the rebellion since it was the first thing I did after winning renown from tournaments. None of the other faction's lords would join my rebellion until I had captured two towns and several castles and it was very difficult holding onto all of the territory by myself. I also had trouble with a second faction that declared war on my rebellion. This happened because I captured territory that the faction I was rebelling against (the Sarranid Sultanate) had stolen from the Kingdom of Rhodoks. This meant my first castle actually belonged to the Kingdom of Rhodoks and they declared war against me instead of the Sarranid Sultanate to get it back.

My suggestion would be to first focus on territory that normally belongs to the target faction since you won't have the political standing to make peace with other factions who want their territory back. Also, before starting your rebellion, build up relations with lords (by doing quests and maybe even mercenary work) in the target faction so that they will be more likely to join you. You can also get lords from other factions to join you as well. The easiest lords would be the troublemakers (the high controversy ones that ask you to start wars between their faction and others). As you build up your rebellion, lords (especially the controversial ones) will join you from all of the factions so it should get easier after a while. If you're female and married, your husband should also join you in this rebellion so maybe try marrying one of the target faction's lords before beginning the rebellion.

Also, when you capture towns and castles you decide who gets them. In the beginning, this will just be you and the claimant. If you give it to them, they will slowly garrison troops there but can't actually defend it since they're in your party. I would suggest not giving much to them overall and take most of it (especially towns since they can generate income) for yourself. When you start to bring lords onto your side, give the castles and single villages to them but keep the towns for yourself. This should keep them mostly happy (they will be angry if you don't give them anything) and they will patrol around their territory and defend against enemies.