As I know there are two different ways to rebel in Mount & Blade: Warband:

If king did not give the lands you conquered, you can renounce your oath to the king.And you can create your kingdom easily. This is the easy way of rebellion.

The second way is support the claimants on throne. This guide will be about this way

In Calradia there's a claimant for each kingdom.

• Arwa the Pearled One - Sarranid Sultanate
  
• Dustum Khan - Khegrit Khanate
  
• Lady Isola of Suno - Kingdom of Swadia
  
• Lethwin Far-Seeker - Kingdom of Nords
  
• Lord Kastor of Veluca - Kingdom of Rhodoks
  
• Prince Valdym the Bas[t]ard - Kingdom of Vaegirs

İf you want to learn where these claimants you need to find traveller and pay 30 denars.( you can find them in tavern)

When you decided which claimants to support. You will oath to claimants like a real king. And you will become a rebel

(You need at least 200 renown for talk with claimants)

When the rebellion begins, you will find yourself surrounded by dozens of lords. You can't fight lords with recruits. Because of that your army should be well-armed and your economical situation can afford your defeats. (my suggestion is don't start rebellion before you make 10-15k money(at least))

You should be in a good relationship with lords. You will understand what the lords want, then you will try to persuade the lords to join your side. This stage is more peacefully than others. Because you shouldn't fight with them when your side weak.Basically this stage for preparation for battles.

For example you were a vassal of King Harlaus and you have chosen to rebel. When rebellion begins, you will become marshall. Your faction call as "Swadian Rebels" and your castles or cities look like you created a new kingdom(with red colour).

Fight till the all castles and cities will be yours.
When you conquered a new place share there fairly with lords. Don't forget they can switch sides.
Don't forget to keep relations high with claimants. This is the most important things for your own good.

When rebellion ends your faction will become what kingdom you were part of.
You can't be marshall anymore but if you have enough renown you may be marshall again. Because of that your relations can't worse with vassals.

Guys I'm sorry again for my language. As I said I am not a native speaker.