This isn't optimal for pros, this isn't the highest damage build. This is for the bronze-plat players who want to play Azir but feel like he's too hard. He's not too hard, people just keep building him in the hardest, most unforgiving way possible.

The most popular first item on Azir is nashers tooth. It's a great item, but lets look at what it does for Azir:

attack speed - Attack speed is great on Azir, but what we care about is effective damage. Raw attack speed simply does not offer this unless you play flawlessly.

ability power - Ability power is also great on Azir, but again, we care about effective damage. Raw stats that seem good at first glance don't always translate into effective damage.

cooldown reduction - This is the only thing that makes nashers tooth a situational item if you need cdr to keep up with certain champions (you usually don't)

On hit magic damage - completely wasted on Azir. This doesn't apply to his soldiers and you should never auto attack champions outside of early lane.

Rod of Ages: This item forgives any mistakes you make. Get caught and the health protects you. Use your mana poorly and the insane amount this item gives makes it okay. On top of all of that, it gives 120 ap by the time you are ready to fight. The perfect first item for Azir.

Sorcerer's shoes: You need boots at this point and magic pen is a great stat since Azir has great base damage.

Rylais: This allows you to actually fight. The slow allows you to kite many champions one on one and win team fights easily. The health keeps on stacking, which allows you to E right on top of champions. (something you don't want to do with a squishy Azir). This is Azir's main tool and why he takes awhile to get going.

liandry's: This item can be replaced depending on the game, but for the safest route you want to take this as your 4th item. The damage is doubled because you have rylais, the health keeps stacking, and the magic pen goes nicely with sorcerer's shoes.

Defensive Items: If your team is doing enough damage and you want to be even more of a threat then end with a banshees and a randuins. Once you take out any high damage targets from afar you can E right into the fight to clean up. If you are on the losing side of the fight you and your ult make an amazing front line for your team.

Offensive Items: I don't like ending the build this way but if your team isn't doing the damage they should be then you are forced to build these items.

This build struggles against champions that can roam early and snowball the game. Most top laners take time to reach this level, which allows Azir to farm and stay in the game.

At level 6 you should be saving your TP to use whenever your top laner shows up in a fight down below. Even though you are weak you can still help even out the fight. This doesn't work as well against a mid laner who can roam all they want without using TP.

Early game will be rough. Focus on farm, play safe and play smart. The faster you get rod of ages into rylais, the faster you can actually play the game.

Stay positive. It's getting closer to 20 minutes into the game, you have had a very minimal impact so far and you just got rylais. You know you played well so far even though you have zero kills. You know you just hit the biggest power spike in the game. This is where Azir shines.

Participate in one team fight and you now have the most damage dealt in the game. Participate in two and that's usually the end of it.

E > Q: Late game if you see a squishy enemy get too close, E then Q to them. This extends your range by a large amount and catches people out. A regular Azir would not go in like this because you would have no escapes, but this is not a regular Azir build. Once you are in and have a shield continue to put up soldiers and attack. You will melt any squishy and they can't scratch you due to all of the health. Very few champions can do anything against this aside from a maybe a vayne with an ultimate.

E > Q > R: This is what Azir is most known for, but you can finally do it safely now. With a regular Azir build you are far too squishy to engage, and it takes skill to pull this off without dying. With this build you have plenty of defensive stats to survive. Do it all you want, keep practicing it. One of the most satisfying things in this game is getting a multi person ult into your own tower.

Pros:

Strong late game
Very forgiving and fun way to learn
Great ability to carry low elo games

Cons:

Weak early game
Not always a good replacement for a top lane tank
You will get yelled at for not building the meta nashers build