This will be a quick and swift guide for the top lane Illaoi without any specific instructions in how you play her.

This is the way I've been playing Illaoi at the Gold 1 rank with GREAT success.

Doran's Shield
You want to start with the Dorans Shield because its the most beneficial item for early levels vs pretty much any opponent. Since it blocks 8 damage, which is a lot at level 1-3.

Phage
You want to get Phage as fast as possible since it naturally bulks you up and give you that movement speed when you attack. Since you lack gap closers this is very good for making chases or escaping.

Sheen
Sheen is perfect for your combo with W+AA. After you have finished this you should not rush into Zeal to get a fast Trinity Force since Zeal really doesn't add much to your strength compared to getting the chain vest. At this point you are really strong compared to mostly any opponent since all these items suit you so very well.

Trinity Force
This is the only offensive item you need. Tons of Damage.

Mercury Boots / Ninja Tabis
These are of course situational. If you struggle vs a Fiora in lane you should of course get the Ninja Tabis or if the enemy team is full AD.

Dead Man's Plate
This is the bread and Trinity Force is the butter. Since you can crit with your W the passive from this item + Trinity Force crit chance you will deal massive damage with your W. It's really insane!

Randuin's Omen / Spirit Visage / Thornmail
These are all situational items. If you lane vs a crit-bomb like Yasuo or if enemy AD-Carry is really fed Randuin's Omen is really good since it reduces damage from crits. It's also a good item to activate right after you use your R to keep the enemies in your tentacle-zone.
Spirit Visage is great because it increases the healing from your Q passive. Great choice vs AP-heavy enemy team.
Thornmail is the item I usually go for in current meta because it really makes you a juggernaut with super high damage output.

I usually go for at least 2 of these items. It's the build order that's situational.

W is your main ability with this build since it works really well with Trinity Force and Dead Man's Plate. Harsh Lesson can crit and with Trinity Force and Dead Man's Plate you will crit huge damage.

Gives you great tankyness with 8% bonus healing from your Q passive + the 3% lifesteal evert 4 seconds when you hit an enemy champion. This suits Illaoi really well and is in my opinion the best way to go.

In the Cunning tree you get the biscuits which gives you both mana and health from your potions. You also get the Merciless perk which suits you really well in teamfights when you dash out your R on multiple enemies. The Bandit perk is always very good for top lane because you will earn 10 gold when you hit the enemy champion (every 5 seconds) and it also gives you 1g for every minion you fail to last hit.

Attack Damage / Armor + Scaling Magic Resist because the scaling magic resist gives you more magic resist in the long run and if you are up against an AD-bases champion in top lane it's more beneficial.