I know that captains respawn simply by passing time, or "allowing" a random Uruk to kill you will get them promoted to Captain.
But how do Captains get promoted to Warchief? I spend a while using the "Pass Time" function and while lots of captains were generated, none ever got promoted to Warchief. 4 of my warchiefs are dead so there is room for a captain to just step up.
Best Answer
Captains can get promoted to Warchief when doing "Duel" events.
Here is what a Duel icon looks like and the description of one.
Although the best way to get a new warchief is to accomplish Betrayal missions.
Betrayal Mission
Even at the top ranks, there is always struggle for power. A Warchief’s bodyguard might betray him and damage or outright kill him. If the initial assassination attempt is unsuccessful, the betrayer will have to fight the Warchief and all his other men which would certainly mean his death.
If players choose to step in at this point and kill the Warchief, the betrayer immediately becomes the new Warchief and gains a power level. This is useful for drawing out a Warchief with the plan on killing him especially if one of his bodyguards is already dominated by the players.
(Credits for Betrayal mission)
The new warchief
The new warchief will be one of the captains serving the one that just died.