Iceborn + Glacial Augment = Gap closer, something Illaoi lacks and struggles with. Use W to proc Iceborn and Glacial Augment to slow them, and catch them. You can also E them and keep chasing, so the slow that comes when they leave the radius of the E will slow them even mroe.

Passive + Death's Dance + Visage = Insane Heals. Pretty basic here, but these runes and items give you good sustain.

Passive + Revitalize + Death's Dance + Visage + Sterak's + Ult = Insane 1 v 5. Every time you land a tentacle, you get a nice heal of 5 % of missing health (which is more because of the increased heals from Visage [increases heals by 30 %]. In addition to that passive boost, you get Death's Dance that heals you for 15 % of the damage you deal to targets. you also have Revitalize, which also increases your heals on low health, which work swell with Steraks proc, a shield that's 75 % of your health, with 30 % tenacity/CC resist (which helps you land more tentacles)

Passive + Corrupting Potion + Biscuits + Time Warp Tonic = Good laning phase. Illaoi can and will dominate laning phase. These things just make it even easier to stay in lane for a really long time if you manage your mana correctly.

Laning Phase - Set up tentacles in all 3 bushes, ward smart, and play safe-ish until level 6. Once you hit 6, be looking for an opportunity to E someone. They can do 3 things - Sit there and take it, go outside the circle, or fight you. Once they fight you, ult them when they are near the vessel, spawning and additional tentacle. W them. A lot. Don't forgot to land Q's, and if their vessel is gone, make sure to E again.

Surviving Ganks - If your ultimate is on cool down, it's best to, again, play safe-ish. Once your R is ready, you can fight ganks easily. Once you see the enemy jungle, don't react until they commit to it. Once they commit, run to the middle of the 3 bushes. Once they close in on you, E the one of them. They will come in to fight. Ult when the vessel, JG, and top are near you. Focus on killing your laner, then the JG. You should get both of them. After that, push wave and back in you need to.

Team fights and Late Game - Late game your job is to land E's and whittle people down. In team fights, you might need to Flash + Ult to engage a fight. If you are in a situation where you can E + Ult at least 2 of them, take it if you have backup. This can result in a nice ace, or even pentakill. With my build your ult never has more than a 60 second cool down starting mid game. If you want to split push get Titanic Hydra, ZZ Rot, etc, and if they send people to kill you, E + Ult them and take down as many as you can before death.

I'd just like to add that to be good with Illaoi is all about reading the situation, knowing your limits, and timing. A good Illaoi is an experienced Illaoi.

Have fun, and remember: Ban Fiora every time, because no matter what, she will slaughter all of you late game.