Hello,
I recently picked up Azir and he's so fun to play. Definitely the hardest champion to learn so far from me (I don't play Shaco - Lee - Cassiopeia) but he can pull off really good moves.
Runes:
AP Quints - Mandatory
Mpen Red - AS Red were popular once but now most pro runs Mpen Red (better harass in lane)
Health per level Yellow - Even against a high AD threat (Zed/Talon) I run these, because you are rushing Seeker's anyway. Plus, your base health is really low, so you protect yourself from the jungler/roaming support DMG.
CDR per level blues (my pref) or AP per level - I feel like Azir at 35% CDR with nashors is really really good. If you want to build Morello+Nashor don't get CDR as you'll exceed the 40% cap.
Masteries: Thunderlords no doubt. Don't forget to take Attack Speed instead of 2% DMG. Worth a lot.
You are really safe in lane in general. For hard AD matches, you take a second doran + seeker asap, every game.
Against hard AP matchups (LB, Ahri), get negatron into abyssal or nashors.
Depending on how well you can farm, Morello's is not a necessity anymore; a second doran could help if you really need the regen. If you have a decent or good matchup, i'll always rush the nashors, then Rylai's.
In the laning phase, my usual strategy is to fast push lvl2. ASAP, get 2 soldiers up and Q to harass and proc Thunderlords.
Rinse and repeat. Always use Q aggressively only if you are sure of the ennemy jungler position or you may die.
Sweet combo: If you already have a soldier in place, you can E to him but mid-dash Q to the ennemy. You'll proc the shield and then you can ult and kill him.
Late game you can play behind carries and you will melt probably everything that comes close, or you can go balls deep to make crazy plays. Make sure to have a Zhonya for this kind of play. And be prepared like me to miss a lot of these plays at the beginning. Small tip: don't press R too fast when you just E'd, or you'll push them forward instead of backwards to turret/allies. Another small tip: Azir in small 2v2 and 3v3 is REALLY good early game.
Hope it helps.