Hey there!
My Name is Nio and I play Illaoi for some time now, especially as a ranked onetrick pony.

Also on the PBE-Client.
Illaoi is a way too strong Champ, if you know how to play her.
The Most important thing in my Opinion is her E (Test of Spirit) because you can do very much Damage with this Spell. If you hit your E you win your Lane. If you dont, you are still strong but may not win your Lane properly.

Choose between Unflinching and Demolish (Slow Resist vs. nice push on Turrets)
As well as between Overgrowth and Revitalize, I guess Obvergrowth is stronger but Revitalize heals you more while you ult and that can save your life.

I experienced Black Cleaver as the best Start because of the Slow Effect. I tried to start with Sterak's Gage a few times, but the effect isn't as strong in my opinion. Still worth a try. After the first Item i would recommend buying some Tankiness, if you are not completely fed (buy Offensive then - Sterak's Gage/Titanic Hydra to dominate everyone). Vs. a mainly AP team i would recommend Spirit Visage because you need the heal effect, especially while you ult, but Abyssal Mask (because of the Mana) is also a nice choice. Vs. AD i would recommend Frozen Heart, because you get much Mana and it works great with Spirit Visage (so much armor & mana + health & MR).

I prefer the Q first to push the lane and deal some damage, but i guess taking E first works as well. Max your E first because you need the percental damage on your Enemy, to get him away from your Farm. Ideal Ult is between your enemy (or his team) and the ripped Soul, because you get one extra Tentacle and all the tentacles do more damage on him when hitting the champion & the soul. When the teamfight starts hit your E, go in, ult, and slap everyone.
Standard Lane Combo: E -> Q (hit champion + soul!) -> W
Teamfight: E -> move between the soul and the enemy(-ies) -> R -> W -> Q (W: 2 sec coolddown -> Slap them fast!)

When you hit your E your Enemy has almost no Chance to fight you because you can slap him hard with Q (ideal: on Enemy and Soul) and W. Always wait for the ult until you have a nice position + E ripped out (ideal: the ADC, AP carry midlaner).
Problem: If you dont hit your E you are quiet weak in teamfight and your ult only works on 2-3 people well. It is still quiet dangerous then.

Dont go in solo and engage a fight, because you are most likely not tanky enough for that. If you have a nice position it is still possible that you win 1v5 E-R-W-Q-W combo.
When you dont hit your E you are in trouble because your normal on-hit damage isn't so much normally. Especially versus fast moving champs (Yas, Lucian, Riven, etc., even Garen) you have to focus completely on your E, vs. slow moving Enemies (Nasus, Yorick) your Tentacles are way more likely to hit.