Rush a codex, buy a blasting wand, finish nashor's with the Codex, turn the Blasting wand into a Rylai's. Reason I do it in this order is because while Nashor's is GREAT, it offers no damage past the codex, so I buy a Blasting Wand early to give me some AP to keep my trades in lane in my favor. After that I buy Sorc Boots, reason I do this is because I use CDR scaling glyphs for Azir, overall late game this gives me more dps than ionian boots and since Azir's lategame is where he shines, well, yeah.
Past that I buy Rabadons or Zhonya's next, depending on the enemy team and if I'm doing good or not, then Void Staff, then buy whichever I didn't buy earlier (rabadons or zhonya's).
If I get a lot of kills early on I might pick up a Mejai's, Azir's super safe if he's ahead so it can be cost effective and can easily replace your void staff or even your boots. But I wouldn't do it in a ranked match of course.