Hello! And welcome to my Azir build, where I attempt to distill whatever knowledge I have accumulated about this champ from several years of try-harding. I hope you find it helpful!
Disclaimer: I am a casual player. I don't play ranked. I am personally ok with this.

That being said, I still have about 700K mastery with Azir, and have tried every build under the sun. I know some things - perhaps not as much as the high elo Azirs with 1 million mastery points, but some useful things nonetheless.

That being said, this build is what I'm currently using, and have found success (and, more importantly, ENJOYMENT) with. Azir is a champion that has fallen into a stagnant build pattern, and has seen little to no variety. Nashor's-Rylai's has been his core for an eternity, and this season I've been trying to figure out some alternatives. I'll be sharing with you my current favorite.

I'm back to using standard Azir runes, the traditional scaling CDR build with Pen, scaling health, and AP. This gives you relevant Azir stats while building RoA. It seems like RoA is a poor choice, right? Lacking CDR, attack speed, pen, etc. However, I have been finding that even the best Azirs are generally too squishy AF to be able to be effective in the mid and late games. And since a lot of things out-damage him these days earlier in the game, it's easy to lose fights that seem winnable at first glance.

This page gives you everything you need to make sure you scale effectively with these items. You can conceivably switch out the Pen reds for attack speed reds, and maybe rush your Sorc Shoes a bit earlier, but I don't really recommend this. It's better to keep the early pen, in my opinion, and just build Beserker's Grieves instead of Sorcs if you want more AS.

People these days are trying builds with no CDR in runes. I'm old-fashioned. Azir feels terrible to me without this stat. Anywhere around 30%+ CDR is good for Azir; any less and he feels clunky and slow to me. CDR somewhere in your rune page means you don't have to worry about buying some rinky-dink 10% CDR item (Zhonya's, Banshee's) just to make him feel more usable.

Call me crazy, but I'm currently in love with DFT on Azir. I like that it essentially empowers every auto, instead of just one out of three, or whatever. Makes early trades in lane no less significant than they are with Thunderlord's, and you're less pressed to land three consecutive attacks. The scaling on DFT feels quite good to me as well, and the extra MPen from Piercing Thoughts comes very much in handy against tanks. It also gets more bang for your buck if you buy Sorc Shoes or Liandry's. Helps to make Void Staff less a required item for every game, and more of a response to teams with tons of MR stacking.

You can conceivably take Meditation in the Intelligence tree with this build, if you really think you want the extra sustain, but now that your kill pressure actually goes down significantly without this mastery, and that RoA gives you all the sustain you should need for the rest of the game, especially in conjunction with Blue Buff whenever possible.

You could also take Battle Trance instead of Double-Edged Sword, but I prefer the consistent increase in DPS, especially with this tankish build.

One of Azir's biggest issues (for me, at least) is his lack of sustainability. I have hated the tyranny of Nashor's first item for as long as it's been a thing (so like, forever). Every other mage gets to build an item that allows them to actually use their spells - BUT NOT AZIR. Well, I disagree. Azir, the ultimate control mage, can't control much of anything without Blue Buff, it seems, and I think the emperor deserves better than that. With RoA you can fix all that, and then not get instantly blown up by the roaming bot lane, or the failed Shurima Shuffle, or by teamfight mispositioning. It makes Azir a bit more forgivable, really without sacrificing damage or usefulness, and gives him more of a window to react to surprise engages and chance opportunities. And as a champion that does need to dive into the thick of things in order to carry effectively, an item like RoA can offer you more than you might think in terms of stats and passives that allow you to do this successfully with far fewer (if any) deaths.

From there you do still want to build Nashor's, because the item is still excellent on Azir. You could go Attack Speed reds and just buy Grieves and get the same amount of AS as Tooth, but then you lose a lot in damage from the loss of Pen. Keep in mind that your Piercing Thoughts mastery requires at least some flat MPen to work: 7% of 0 = 0. These two items, RoA and Tooth, constitute your only core with this build.

From here you have a couple of options, based on situation and boot choice. If you opt for more AS with Grieves you'll want to get pen first with Liandry's next - WHICH will give you an amazing SAME AMOUNT OF HP AND MORE AP THAN RYLAIS. Literally my new favorite item on Azir, especially with DFT. If you went Sorc Shoes you might want to go either Rylai's for even more tankiness and utility, or Luden's for more damage. I slightly prefer the Luden's, but tank Azir can feel unbelievably strong with this build. Again, pick based on preference and/or situation.

Deathcap is always an option if you're snowballing or aren't doing any damage. You can get it either third or fourth, but I would try to get it into your build somewhere, unless you're still effective somewhere around 400 AP. I don't want to say this item is required for carrying, but sometimes it's required for carrying...

Since the mage item rework I've been really underwhelmed by Zhonya's Hourglass. You only need this item with this build if you're against an assassin like Zed, who will find a way to easily cut through your hefty HP. Banshee's feels similarly to me - I'd rather have old Abyssal, but if you're against someone who's able to still burst you down (Veigar) you might still want this item. Fortunately they both offer some meaningful benefits, and don't synnergize poorly with a tanky Azir.

I like scaling W first, unless the opponent is highly mobile. You can go back and forth between these skills as you need, choosing to put points into W when you want more attack speed and Q when you want more ability to reposition and burst damage. The extra AS scaling through your W makes your farming easier, procs your DFT more, and helps him feel less clunky while you build your RoA.

** Always prioritize your ultimate at the appropriate levels. **

Barrier is all-around the best spell, still. Viable alternatives, however, now include (in order of importance):
Heal: Good against other sustained damage champs who don't have Ignite
Exhaust: Anti-assassin, anti-dive
Teleport: Great sustainability, opportunities to create map plays, allows you to make good
use of your powerful wave-clear
Cleanse: Anti-CC
Ignite: Extra early kill pressure. Doesn't matter much late game.
Ghost: For high mobility champs, high CC champs, etc.

Azir is predominantly a farming champion. Prioritize farming early, but also use your range to poke and harass constantly. It's easy for you to bully most champs out of lane, but don't necessarily make that your goal. Instead, focus on CS, river vision, and light roaming to help your jungler get control or get out of danger. You can sometimes pick up some kills this way and then snowball your lane.

Once you have RoA, you'll want to then start playing more aggressively, taking opportunities to roam and take objectives. Constantly beat up on your lane opponent whenever they show their face. With RoA it is disgustingly easy to poke out and out-sustain most opponents, so that they have to go back and you still have resources to roam Top. One of Azir's biggest weaknesses is that his roaming is less than stellar, even after finishing an item. I have found, however, that the RoA build actually makes his roaming around 15 minutes more viable. Whatever you do, take advantage of your easy lane pressure to maintain vision of river and enemy jungle entrances. Use your incredibly malleable dash to adapt to various situations and either get out or turn it around with an ult.

Your goal is to dish out as much damage as possible from the back line, but with tankier stats you can move up to do so much easier. Engage with your ultimate when you see an opportunity to either catch someone out, or to put a group of people in an awkward position. Plop a soldier down wherever they land and let it get to work dealing AoE dmg. RoA makes your engages more reliable and safe, and therefore more effective. You can't do any AoE from a gray screen ^_^. You are a huge component of your teams DPS and engage potential. If you waste either of them the enemy will win the fight, unless your other teammates are all doing really well.

If you've played your cards right (meaning you haven't fallen behind in CS, have at least two items, and have several RoA stacks) you should be a force to be reckoned with at least by 25-30 minutes. Your best bet is to sit midlane, set up shop, and shove like mad with your team working around you. This way you're in a good position to contest objectives and be around for teamfights that break out. You can, however, conceivably become a split push nightmare with your sustain, mobility, turret passive, etc. Keep this in mind as you roam to unmanned lanes in need of pushing, which you should be doing at this point often. As your team's main source of wave clear, and as a champ that needs a lot of farm, you should be moving to lanes with big waves whenever you can.

In lategame teamfights you want to be careful. If you die - and it certainly is still possible despite your HP pool - your chances of losing the game increase exponentially. Generally poke them out a bit first. If they start to retreat follow up with a shuffle if you can get a good angle or just chase them with E. If you took either Rylai's or Luden's chasing should be easy for you. If someone dives you, ult them away and get as much damage onto them as possible. Your wall only lasts 3 seconds at all ranks (as an aside, this was the most ridiculous nerf Azir received. Completely unnecessary and just totally castrated his zoning potential, which is sort of what Azir is all about, c'mon Riot, plz), so if you don't deal with the threat that dove you within those 3 seconds you will likely have to deal with them immediately after the wall is down when their cooldowns are likely up. Your wall is your biggest strength and biggest weakness. If you use it correctly you can turn fights and engages and win them easily; if you mess it up in any way you are likely going to just get CC'd and die. Yes, your RoA makes all this easier, more reliable, and safer. However, you are not totally invincible.

At the same time, you aren't going to get much better at the Shuffle if you never use it. So don't be completely afraid of engaging with it if you think it could make a play. You are building a hefty version of Azir, after all.

Last tip, don't be bottom lane when Baron is up ^_^.