Illaoi is an Offtank, which is brilliant in Teamfights.
Your Sustain and your damage in Teamfights is just insane.
So you should force Teamfights as quick as possible.
But if its needed, you can also Splitpush very well, cause you´re a sick duelist and you have a good waveclear.
You should persuade your Team to get Objectives (Nash, Dragons and herold)
This Spots are lovely for Teamfights with Illaoi and the enemyteam won´t have fun to contest these objectives.

Illaoi is one, maybe the choosen one, of the best duelist´s.
If there are some tentacles near, your damage, and also your sustain is insane.
Illaoi is the Champion who is maybe the hardest to gank.
In higher Elo, the Jungler´s will know, so you will have a relaxed time on lane.
In low Elo, the Jungler´s will hopefully don´t know. Yes, Hopefully!
Don´t worry about getting ganked (You should ward anyway) because when you have your Ultimate up, you will get probably a double kill.
You should use your E-ability as often as you can (but be carefully with ur Mana)
Its an easy and safe way to poke your enemy and being a Vessel will drive ´em crazy.

If you want to all in your lane enemy, or you´re forced to fight the order of your abilitys is very important, but not very difficult.
Make sure, you are near to your, by passive spawned, tentacles, so your impact is ways higher.

At first hit your E-ability on the highest-priority target (Adc or Apc)
Now use your Ult instant and hit your W-Ability on your Enemy (not his soul), so he gets slowed by your Iceborn-Gauntlet and all your tentacles can rape him easily.
After that you use your Q-ability and make sure, that you hit as much as possible enemys and the soul you drained out with your E-ability too.

That should be enough to destroy weak targets, even with this defensive Itembuild you have a sick burstdamage. The more enemys you fight, the higher is your burst damage.
Against tanky Enemys try to bind them on you, and hit both, the enemy and his soul.

Illaoi has a welldone Championdesign *Irony off*
Illaoi just got a little of weaks.
She isn´t very mobile, so you should buy boots early, to dodge spells.
Don´t try to chase mobile champions, if they´re not infected.
To compensate this, i choosed the Iceborn-Gauntlets for my build to bind them on you.

Befor Level 6 you shouldn´t overestimate your sustain.
And before you have Black Cleaver, you shouldn´t let your enemy to push too hard, because farming under tower is a bit difficult.

In Fights the best counter against you, is to get out of your ultimate.
So mobile champions are hard for you. Also ranged champions can poke and kite you easy
in laningphase if you´re not carefull enough. The worst about Ranged Champions as enemy is, that if they´re carefully by them self, they can hit you while they don´t get into the range of your Ultimate.

Flash is absolute important for you, to cope with your immobility.

Why I prefer Exhaust: Its strong in 1v1 situation and helps you to bind your enemy in your tentacles range. Also this Summonerspell won´t get weaker with the time, rather it becomes stronger. Also your team will thank you, because you can have two exhaust´s in your team, Your enemys will hate you for this!

Ignite:
Ignite is useful against enemys, which have also a strong sustain. And it helps you to get early kills, so you can snowball faster.

Teleport:
I don´t prefer Teleport, but if you like it, it fits also.
In my opinion the time, you need to travel is to high for effective TP-plays.