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Conqueror got nerfed!

Buff part: Conqueror stacks from ALL SOURCES of damage, making you an animal with any champ. Nerf part: The starting damage was 3-15 / stack, this made you an insane early game champ with almost any champ (Azir included). But it was nerfed down to 2-6 / stac, and buffed back to 2-10/stack. Conclusion: I'd say it's still great after the 2-10 Buff and the all sources.

Aggressive Poking:

Try to farm with your W, but hit the enemy champion at the same time. Try to Poke the enemy with 1-2 soldiers and Q, while maintaining a long attack range. ALWAYS.. ALWAYS WARD YOUR BUSHES. If the enemy jungle started on top side, try to position yourself in the bottom side of mid-lane, this way he/she will have a harder time ganking you. Although, your main priority is poking the enemy, try to last hit as mutch minions as possible, while you don't let the enemy farm. Why? This way the enemy will fear you and let you farm the minions for a while. This will make sure that your early gold and item advantage will become a thing.

Level 2 POWER SPIKE:

Try to play around the level 2 power spike, when you get to level 2, get your Q as the second ability. Put down 2 soldiers[W] in a way that you can hit the enemy champion, and when he/she moves out of your Soldier range, press Your Q in front of Him/her and auto attack as hard as you can, if th enemy is on low hp: Flash and ignite may get the First blood in your hand.

LEVEL 6 POWER SPIKE:

Before You reach Level 6, make sure that you are under your turret and let your enemy push the lane a bit.
When you Finally hit level 6, Put down a soldier[W], start dashing towards it[E] and press [q] while your dash is still not finished and try to get behind the enemy this way, if your ult can reach the enemy , then press it towards your tower, put a W under your enemy and start attacking. IF the ult is not in range, use a flash before using it. (use Ignite aswell if its needed)

1ST ITEM POWER SPIKE: (Liandry's Torment)

When you get your first item, you'll have a mini-ignite (about 10% as effective, but has NO COOLDOWN). At this point buy a control ward, ward one of your bushes with that and the other with the usual ward, ths way you'll be harder to gank.

Put down 2-3 soldiers and Aggressively poke the enemy as hard as you can (now farm doesnt matter that mutch). When the enemy is on low hp, Drift (W -E and press Q while still dodging) behind the enemy, if your ult range isnt in range, flash too and then press your ult towards your base. Put down an ignite aswell and this should do the magic. ;)

Why?

Since you have basically a Mini ignite, you can deal Massive amounts of damage with just aggressively poking.

Main Flash/Ignite Combo

Flash is almost neccessary for any champion out there, because of the escape potential and enemies may not escape if you have it.
Ignite is for more aggressive plays, you place in your soldiers, attack from far away and if the enemy wants to escape, you press E and then put an ignite on the enemy, if everything worked fine, you have a kill.

Alt Flash/Barrier combo

Flash is almost neccessary for any champion out there, because of the escape potential and enemies may not escape if you have it.
Barrier is for more defensive plays, or if you want you can go balls-deep and it can save your *** from dying.

Starting items:
Amplifying Tome x1 (435G)
Health potion x1 (50G)
Total: 485G
Amplifyling Tome: Gives you 20 AP and the Starting item of Liandry's Torment
Health Potion: It's always good to have at the start, might save your life.

Core items:
-Liandry's Torment (3100G)
-Rabadon's Deathcap (3600G)

Liandry's Torment: Gives 75 AP and 300 Health.
Passives:
-Madness: Damaging champions increases your damage dealt by 2% for each second in combat (Stacks up to 5 times)
(MAX: 10%)
-Torment: Dealing damage with a spell on an enemy deals 1.5% damage of their max health per second for 3 seconds (4.5% of their max health) If they are Movement impared(Stun,slow,terrify ect..) it deals 2.5% per second (7.5% of their max health)

Rabadon's Deathcap: Gives you 120 AP.
Passive: Increases ability power by 40%.

Main (Precision):
-Conqueror
-Triumph
-Legend: Alacrity
-Coup De Grace
Secondary (Sorcery):
-Manaflow Band
-Absolute Focus/Scorch

Conqueror: The Heart of the build, gives you massive amounts of adaptive focus and stacks with each soldier attack, basically, the longer you fight, the stronger you'll be. (Max : 50 Adaptive Force (50 AP))

Triumph: Can save you in lots of scenarios, mainly in teamfights, but when the enemy jungler ganks you and you kill one of the enemies, it may restore enugh health to either kill the other one aswell or escape.

Legend: Alacrity: Since Azir's Base Attack Speed Growth is increased from 1.5% to 3% + the soldiers give bonus AS aswell, you don't have to buy Nashor's, instead just use this rune for your basic AS source.

Coup de grace: No explonation, just more dmg :D

Manaflow Band: One of the most important runes in this build, it keeps you on the lane because you get mana for each champion hit (has cooldown) and after you reached the 250 mana (10 times the attack happened) you'll get your missing mana's 1% every 5 seconds.

Absolute focus: Basically, more damage aswell, can be swapped out for Scorch for more Damage-Overtime effect.