Basically, you need to take advantage of the gold lead you get. Illaoi is amazing by herself in lower elos (I'd say about anything under plat 3 and she typically will work fine). You need to split push, and do so hard. Take towers, make them have to fight you 1v2 or more. This gives your team the advantage everywhere else on the map since they have less enemies to deal with.
Make your intentions clear to your team. If you kill their laner and your jungler is near top side, get them to do herald with you. This makes fighting illaoi hell for nearly twenty minutes.
Make use of wards. Don't be afraid of ganks, but do be prepared for them. You want ganks to happen, and you want the upper hand with your E + R. Take advantage of this, and the enemy will hate trying to fight you as most people do not actually react to her combo fast enough and will die.
If you can't take inhibitor turrets, you have one of two options. Either split push bot lane and take over those towers, or it's time to group. If you ARE forced to group, do not be the one to try to engage if at all possible. You can whittle them down by landing your E (and they will either back off or go in on your team which is good for you) but don't be the one to engage. If you absolutely must engage, only do so when your flash, e, and R, are all ready. Hit R key, and mid ult flash ontop of them. This will catch most players off guard as you follow up with an E and W spam. Target anyone you can who you think will be more likely to eat tentacle hits. Gunning down the fed ADC ia going to be nearly impossible for you in most cases.
Anyway, she really is about objectives and forcing the game to be an advantage for your other lanes by pure dominance. Hope this helps.