Each time you kill an Entranced Pilgrim of Flame a floating ball of fire will travel towards the altar and light a brazier. After a short while the flame flickers out.
You need to get all five braziers on the altar have a fire in them at the same time. In other words, you need to kill at least five Pilgrims, and the flame from the last one must reach the altar before the fire from the first Pilgrim is going out. You also need to stand relatively close to the altar to get the update.