So much of the advice here is honestly pretty terrible. The comment going over what abilities do wasn't even correct FFS.
With Illaoi, you need to understand that your combos and abilities all take a ton of time to go off, more so than literally any other champion in the entire game. Your E will self CC you for over a second, your Q, and any tentacle slams all take an entire second before the slam does damage. As a result, simply E->Q will take a little over 2 full seconds to finish. The ideal combo however is E->W, move to cancel the wind-down of the auto->Q which takes 2.25-2.5 seconds to complete. IF you have your ult up, and there are multiple other foes in range, its instead E->R->W->Q, which will take almost 3 full seconds to complete, but will cause all of your tentacles to slam the spirit 2 full times, plus your Q, at which point your W will be almost off coolldown again for another full slam, at which point your spirit will direct another slam for a fourth slam in about 4 seconds of time, of which all 4 slams happened in the latter 2 seconds of said 4 second interval.
Now, understanding this combo, how much time it takes, and when the damage goes off is very important. Since there is a slam going off roughly every half second in the latter phase, your healing based on %missing HP will be MASSIVE. Unless ignited or inflicted with grievous wounds, if you can survive the first 2-3 seconds, your healing will keep you going for the next 2 seconds, at which point whomever was standing in your ult is likely dead, and other people will have backed off.
So, what you need to focus on then, is surviving for the initial portion of that, such that you can get off your fantastic AoE teamfight damage, and can keep yourself alive with your % missing HP sustain. Illaoi's kit doesn't have the tools to deal with the hyper burst META so unfortunately, you need items to bail you out. The AD, HP, CDR, and passive of
or
make them your go to first item; the former if even or behind, and the latter if ahead. However, they won't provide enough defenses by themselves, so you'll need another more defensive second item, (along with finishing your base boots into either
if they are CC heavy, or
if they are AD/AA heavy). What I have found to be singularly useful for a variety of reasons is
as a second item. Firstly, you have utterly massive base AD which scales into your tentacle, Q, and W damage, as well as increasing the % of damage dealt to spirits that their owner will take, so Sterak's provides a sizable damage buff to every ability but your ult, which is never used for its burst damage anyway. Secondly, it increases your max HP by a good amount, increasing how much you'll heal for at the same HP %, and making you harder to burst; however, most importantly, it provides a massive fuck-off shield which you should be able to time for your massive 4 second combo. This shield alone can be the difference between dying just as you ult, vs surviving the whole combo, and doubling the slam damage. Additionally, its a shield, so your HP will stay at a very low state while its blocking damage, thus ensuring you heal massively from your % missing HP heal for each slam you can get off. Furthermore, it provides a good chunk of tenacity which, when in conjunction with runes and or merc treads, will enable you to shrug off interrupts to your combo.
For a third item, the damage, CDR, passive delay of damage taken, and the extra healing from all physical damage makes
an absurdly powerful Illaoi item. It doesn't provide near as much defense and especially burst defense as Sterak's though, so be sure to pick up
before you go DD.
One other thing is that because you self CC yourself for so long to initiate the teamfight combo, its very useful to first kill a spirit so they have to deal with tentacle disruption in their back line, or better yet, wait for a tank champion to initiate and CC your opponents, then follow up with your E->R->W->Q combo. This will help keep you from being CCed and focused down yourself while you get your lengthy combos and set up off for maximum damage.
Finally, one last note is that you are immune to displacement CCs/effects during the cast time of your ult. This can be especially useful vs some champions to ensure your combo is not interrupted before you can get your tentacles slamming to regen your HP while otherwise CCed.