It's easy as a muslim or crusading catholic. Just build a chapter house/ribat then 'train' 3 crusade/jihad markers and then just destroy the chapter house/ribat. Instant civil war.
As the Orthodox or non crusading catholic the best way to get a crusade going is not to separate your king from the mainland, as that way you will get revolts in many of your provinces. This is not a civil war. The most expedient method is to repeatedly send a few peasant forces into battles they cannot win, then try your loyal generals for treason to reduce their loyalty.
In either case once you're ready, the loyal and disloyal need to be separated into two camps in order to influence who controls which provinces when the faction divides. The king should be in a province on his own or perhaps with some of the other loyalists, as the units in his province won't defect. The loyalists need to be dvided and placed in situations where they can't win or take control of any decent provinces. It's not easy to get it perfectly right but you can do a better job of it than just leaving the AI to it.
I had to do this once with my 60 year old heirless king. Racing against the clock to try to precent my faction from dying out.
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