There are two main ways to deliberately start a civil war within your own faction.
1.) Drive down the loyalty of one of your best generals: Strip him of whatever titles he may have, give him a big army to command, and place him a good distance away from your faction leader. (There's a rough correlation between a general's loyalty and his proximity to the king.)
In addition, losing a few provinces and/or suffering failed Crusades/Jihads can help spur a civil war as well. In fact, you don't even need to go on Crusade/Jihad per se -- just train a couple Crusade/Jihad markers, and then destroy the Chapter House or Ribat that trained them. This will automatically cause the Crusades/Jihads to fail, your faction leader will suffer a massive loss of influence, and you'll very likely then face civil war.
2.) The second way to instigate civil war is to kill your faction leader when he has no direct heirs (usually while he is still pretty young). This method is a little trickier, however, as your faction leader has to have 2 or more "royal generals" running around (most likely his uncles) -- they will then fight each other for the throne after you current king has been killed off.
You need to make sure you do have at least 2 royal uncles when employing this method, though. Otherwise your faction will simply die out, and the game will end. Still, it's great fun; I prefer to start civil wars this way when possible.