Hey,
My IGN is Mumsno, I play in the Eune server and currently ranking as Illaoi #410 in the world (According to LoLSkill).
I play LoL for almost 2 years (Currently I am Platinum IV), and I played a lot of games with Illaoi, and decided to help others to know how to play her, as she's really fun and can carry hard if played correctly.

I know what most of the people think - "Illaoi is an easy champ, just ult and you win", well, no. If you're against low-elo or against people who don't know how to play against her that would be true, but in the high-elo you'll need to control every tentacle and movement, or else you'll get poked and killed. Illaoi is very easy to understand, but hard to master, you'll need to know how to play against ranged champions, and how to ult correctly - a fail ult means death in teamfights.

Pros:
- Easy to learn
- Most of the ganks by the enemy's jungler will be double kill for you.
- Lane bully, can farm easily.
- Carrying HARD if played well, you don't need to be fed.
- Very rewarding ult.

Cons:
- Easily Kited by most adcs.
- missing your ult means death for you/your teammates.
- hard to master
- hard to play in high elo
- You Will be ganked alot, as she's someone to be afraid of.

In the recent Patch (6.11) Illaoi got some buffs:
Tentacle Smash - The New Passive Increases Illaoi's tentacles damage, which is really helpful, therefore it worths maxing it first, simultaneously or second after E.

Important Note: This is just my playstyle, If you do not rely on your E, or not very good with skillshots, you can max Q and then W, For my playstyle this skill order is better but it may not suite yours.

Harsh Lesson - According to my experience, it doesn't worth maxing second, as E and Q are way more important, and you still get alot of damage from W (and the cd doesn't change). It's still not recommended, in my opinion, to max it second. The cd doesn't change, only the damage. And you don't really need this damage to be high early game.

Test Of Spirit - The slow is now buffed (as it's not decaying), I still max it first because it matches my playstyle, but you can max it second after Q, still very strong.

Playing against Illaoi can be easy if the Illaoi is not a skilled one.
You must do the following things:
- If you can, dodge Illaoi's Q, as it heals her and deals mediocore damage early game.
- Try to focus her tentacles, she's alot weaker with out them, BUT NOT USELESS.
- If Illaoi catches you with her E - Test Of Spirit, attack her if you can. It will make the ghost disappear. If you're already a Vessel, kill 3 tentacles and you're out. (This will probably be patched in the next cycle).
- If Illaoi caught you with her E, and you can't attack her - It's better for you to go out of range and receive the Vessel debuff. This way you'll not receive the damage transferred from her E.
- You can try bait Illaoi's ult by being close to her and then flashing ult, but it will not work against a skilled Illaoi.
- Try to poke her as much as you can. Dodge her tentacles if you can.

I Switched to full AD runes instead of CDR, as I still have the cdr I want late game and early game isn't so bad with the recent changes to Illaoi and her cds & damage.

The current rune set include pretty good defensive stats (magic resist and armor) and AD.

These runes cover up the stats Illaoi needs - Armor, Magic Resist, Attack Damage. The CDR she needs she'll get from items, as her current cooldowns are quite low.

• Prophet Of An Elder God: Illaoi's Passive and the origin of all of her skills - every few seconds she spawns a tentacle nearby her (it will be attached to the closest wall to her, if there are no tentacles on the wall in a close area). The tentacles can be controlled with Illaoi's w (Harsh Lesson - ) and will be swinged automatically at any spirits she caught with her e ( Test Of Spirit). Unless Illaoi uses her ultimate, the tentacles can be destroyed, enemys will often focus on destryong these, as it cuts Illaoi's damage massively.
  
• Tentacle Smash: It's passive is the most important thing on Illaoi - every tentacle that hits an enemy champion will heal Illaoi for 5% of her missing health, Including her passive tentacles and her Q ability.
  
• Harsh Lesson: Illaoi's w resets her basic attack timer, and provides a short dash to her basic attack. When Illaoi hits her w, any nearby and in-range tentacles will hit the targets she leaps into. This ability is critical to controlling her tentacles.
  Why should I max it second, and when should I max it last? well, your ideal first skill to max, as I said, is E - Test Of Spirit, as it will help you poke and will deal alot of damage when used correctly with your ult. If you see that your lane tentacles aren't destroyed, you should max W after E, as you can control your tentacles better and deal alot of damage, and also have good sustain. If you see that you often stay without tentacles in-lane (say, the enemy is focusing them), you should max q second, as it's your only source of sustain and long-range poke without tentacles.
  
• Test Of Spirit: Illaoi's e. Using this ability, she can grab an enemy champion soul and hit it, transfering % of the damage to the real champion.
  Important Note: every champion-related abilities, such as Soraka's q (healing from champions) will work on the spirit Illaoi caught with her e, this could be really viable for her team and for herself if she has Grasp Of The Undying mastery.
  It is the most important skill of Illaoi after her q, in my opinion. This is Illaoi's tentacle spawner, and main poke ability. It can deal huge damage to enemy champs if maxed, and if she grab someone and then ults, the damage of her ult will be much higher than just ulting.
  
  Illaoi's e synergies with her ult - If you grab a champion with e, her ult will count it as a champ and will spawn another tentacle.
  Every Spirit Illaoi catch with e will be hit automatically by her tentacles, meaning that if you ult after you caught a spirit, the ult will hit it once with all the tentacles.
  
  Try to avoid ulting without catching someone with ur e, as the ult will be much weaker without ur e. Although there are times when you need to ult without ur e, but you'll need to learn those as you play Illaoi :)
  
• Leap Of Faith: Illaoi ult, max it when you can. Illaoi jumps into the air and then slams the ground, dealing decent damage, but this damage isn't her ult main idea.
  
  For each enemy champion (And Spirits Caught With Her E!) caught in Illaoi's ult range, Illaoi will summon a tentacle. the more champion in ur ult => the stronger it is.
  Again, try to always ult when you caught someone's spirit, or else you'll lose one free hit with your tentacles, and an extra tentacle for your ult.
  
  While ulting, your tentacles become untargetable (meaning they cannot be killed) and your w's cooldown will become 2 seconds.
  

Update: According to Patch 6.10 notes, Illaoi's W - Harsh Lesson Cooldown is 4 secs at all ranks, therefore it's not worth maxing anymore as the damage isn't that important. Maxing Q second is better now, in my opinion.

Most Of The Time I take teleport, as you need to go back to your lane and push it. Against some matchups (like trundle, Riven, Fiora, Rengar) I take exhaust in order to be sure that i'll win trades. Against Nasus I reccomend ignite, but against mundo and vlad take tp and buy executioner calling.

If you know that the enemy jungle will gank you alot (an easy jungler like fiddle will mostly do that), take exhaust - so when he comes you'll do a double kill, or atleast you could prevent your lane from doing the kill.

Farming is fairly easy, as you have an auto-attack reset (w) and you can push your lane easily with your q and tentacles. Push only when you're sure you can handle more enemies coming, or when you have wards.

Illaoi's kit is pretty unique, you need to always be sure that there are tentacles around you, and spawn them when you can.

Early Game:
At the start of the game, you should spawn tentacles around your lane. Start with your turret (incase your lane will be pushed), and then at the walls near your turret and to the bushes. Spawning tentacles is really important as it will help you to farm and poke with ur w and e.
Poke with ur q when you can, but don't do so carelessly as you'll lose alot of mana. Hit with ur e as much as you can as it's the main ability to spawn tentacles and poke.

If you grab someone with E, the tentacles will hit him automatically, so if you e + ult, you'll get an automatic first hit by your tentacles, which deals alot more damage then just ulting. Additionaly, your e will count as a champ, and will spawn an extra tentacle when you ult.

Mid-Late Game:
You should split-push if your quite fed, or if you think that you can go against more then one enemy at a time. If your lane is already pushed and you can't push further - just go help your teammates, one good e will surely help some teammates in need.

TeamFights:
In teamfights, Illaoi cannot engage. She doesn't have any cc ability and she gets easily kited without a slowing item or landing her e, so she needs to rely on others with engaging. BUT, she can grab people with her e - Grab anyone you can with your e (it's preferred that you'll grab the other team's carry, but if you can't reach them - grabbing another teammate is good than nothing). The damage transfered to the target is huge mid-late game, it will heal you and it will help you spawn tentacles around their team.

Of course if someone in your team engage or if the enemy's team engaged - you should E and ult as much people as you can in order to deal the most damage you can. Focus the adc if possible, if not - peel for your team.

Illaoi's e is her main ability which I max first, as it's her main tentacle spawn ability around enmies, her ideal poke and the source of her ult's damage.

Controling your e means controling your tentacles and your ult, and it can also give you a huge advantage over your enemy laner.

1. Vision: Grabbing someone with your e gives you vision of them, even if they're stealthed! Which means if you grab Vayne, Rengar, Teemo and such during or before they got into stealth, you'll earn a long vision on them.
2. Free Hits With Your Tentacles: Nearby tentacles will automatically hit the ghost you grab with your e, meaning one free hit from the tentacles, so you should do the following combo: E - (let the passive tentacle hit) - W, which will triger the tentacles to hit twice.
3. Double Damage: If you grab someone with your e, and then Q their spirit AND their actual champ - the damage from the q will be doubled (not exactly doubled, but it will deal more than just q'ing) as the Q will hit the actual champ, and also their spirit - transfering some percent of the Q's damage to the champ.
4. Grasp Of The Undying: Your spirit counts as a champ for everything, so you can proc Grasp Of The Undying key mastery on it and steal health.
5. An Extra Tentacle For Free: As I said, the spirit counts as a champion, which means your ult will spawn an extra tentacle if you catch the spirit and the champ itself in your ult's range.

Your e is a very strong ability, but if you miss it, or if it's on cooldown, you'll be left vulnerable. If you miss it you should step back unless the other enemy is really low, or if you think you can kill him without it. That's why you are maxing E - to lower it's cooldown and make you less vulnerable.

Here's a short gif showing a nice trick with Illaoi's e:
If your E Spirit's health is higher than the real champion's health, you can kill with it!

Placing Tentacles:
Play Video
Don't underestimate your ult! even though you're low, you can kill ;)
If you killed the ghost of your opponents, you don't need to wait to pull someone in order to ult, as it will attack automatically the cursed target. Here's an example (And also a nice ult flash <: )

just some 1 vs 5s (ult + flash is very useful):


More To come :)

I hope you liked my tutorial, let me know what you think! I'll continue to work on this guide and improve it by your advices.