Illaoi's not too bad. If you're always trying to be real aggressive, you'll probably lose to her though. Her E (soul grab) doesn't go through minions, so keep that in mind. If she does grab your soul, just walk out of range as quickly as you can without using Flash. After 6, if she jumps at you or your spirit using her W, she might be setting up for her ult; watch for her to jump into the air. If she uses it, don't try to fight her, just back off. The main thing with Illaoi is that she heals for 5% of her missing health every time a tentacle hits one or more champs. But her tentacles will only attack you if she leaps to you with W or lands her E, and are otherwise not threatening. While her ult is active (8 seconds), her W cooldown is halved (from 4s to 2s with 0% CDR), so just leave. Damaging her while your spirit is out from E used to reduce its duration, but it no longer does, so just leave if her E hits you. She has no way to keep you around.
You can cut down very significantly on her damage and sustain by simply dodging her tentacle hits. If she uses E and fails to land it, she's very vulnerable for at least ten seconds, and that's the best time to put pressure on her. Run around her to dodge her Q, watch for her ult if it's up, and you should easily beat her in a fight. She should really only ever beat you if she lands her E, so don't let her.