Azir its a very particular champ, I'm still exploring his possibilities.
He is full of ap-scaling abilities, he got no sustain at all and he needs the attack speed to maximise the W (soldiers!) potential.

Pros

• Not a very mana abuser
• Nice mobility
• Teamwork friendly abilities (Passive Turret, Ultimate...)
• Good for sieging or for turtling
• Nice poke possibilities
• Good at splitpushing

Cons

• Hard to master
• No sustain
• Bugged at the moment

The runes of my guide seems a bit weird but give them a try and u will be satisfied!

Azir needs the attack speed to maximise the W power and without the AS runes u will have A LOT less DPS, really precious to:

• Farming early game with ur soldiers;
• Poking the enemy: here the AS runes really shine! The soldier will attack him multiple times in early;

The CDRxLvL Rune its here to cap the CDR with Nashor and Elixir, thanks to this rune you can buy other boots than Ionians (A really good choice for Azir due to the amazing CDR bonus), and go for Sorcerers (Or Bersekers?! O_O)

The rest of the runes are pretty standard, I personally prefer a mix of xLvL/Flat for the Glyphs and the HPxLvL Seals procs great with the E Shield.

Pretty standard masteries, the 9 points on the utility tree gives a bit of sustain (mana based), and mana regen for the early stage of the game.

On the Offensive tree the Arcane Blade its core because gives to Azir better farming and pushing potential (damage to turret!) and its a good sinergy with Nashor.

Attack Speed really helps Azir's Damage (Nashor's Tooth)

HP gives to Azir more survivability and improves his E shield (Rod of Ages, Mask and Rylai)

Azir's Soldiers attacks procs Mask and Rylai effects!

Seraph Embrace is here due to the mana based build and to have fun spamming your mid-low-cd skills.

Rabadon is not present on my full build example because I actually dont miss the bonus it gives. Maybe if you are really fed you can consider to build it early to