Talk to Alexander on the Mountaintops of the Giants by summoning him for the Fire Giant fight, then speaking to him afterward if he appears there. That’s the key Mountaintops step before his final appearance in Crumbling Farum Azula.

Where he is

Alexander’s Mountaintops-related interaction is tied to the Fire Giant arena, not a random overworld spot. If you’ve progressed his quest far enough, he can be used as a cooperator in that boss fight.

What to do

  1. Progress Alexander’s quest through his earlier encounters in Limgrave, Liurnia, and Mt. Gelmir.
  1. Reach the Mountaintops of the Giants and head to the Fire Giant boss area.
  1. Summon Alexander if his sign is available, then beat the Fire Giant.
  1. After the fight, talk to him there to continue the quest.

After Mountaintops

Once that dialogue is done, his final stop is Crumbling Farum Azula, where you can find him near the Dragon Temple Lift area for the duel that ends his questline.

If you want the exact route through his earlier quest steps, I can lay it out in order.