This is just a short guide I threw together to give this build more exposure.
The idea is that your early is gonna suck because without AD wave clear is harder and you wont be able to take long fights until your 2 item power spike.

The core of your kit with AP Zeri is the auto (not Q) when fully charged deals 100% AP in damage. That is literally nuts, this means you will only take short bursty trades in laning phase. Never go for an auto trade without fully charged auto ready.

Your next important ability that works differently as AP Zeri is your R, it has 110% AP in damage. Do you see where I'm going with this? Auto and R alone deal 210% AP. Yeah that's literally AP Twitch levels of damage, unfortunately half of it is in your ult and that means you can typically use it about once every major fight. So far we've established your burst is Charged Auto (CA) into R for 210% AP.

You've just bought Nashors! Your Q deals 20% AP on-hit now. Hm..and E seems to empower your Q as well by 20% so if we add Q to your burst it would appear to be 250% AP oh my. And this is just the burst, if you're a little short your next few Qs still deal 40% additional AP!

You finally finished Lich Bane? Well it's probably not that crazy since you need to use an ability to empower your attack.. What's that? Q counts as an attack AND an ability? You trigger and proc the damage with one button press? ALRIGHT LET'S ADD 50% MORE AP TO THE BURST, landing us at 300% AP oooh the numbers... THE NUMBERS WONT STOP CLIMBING! Your Q will also continue to proc lich EVERY OTHER Q.

Oh I see it now.. AP Zeri may seem very weak at first but once you build Lich Bane things start to look very scary. The amount of damage Zeri can dish out in the first moments of a battle is insanely scary, and might I remind you she can appear almost anywhere on the map with some clever E usage? 300% AP burst is nothing to scoff at and 110% of it is literally AoE, making her a terrifying champion in a teamfight if there is no one to cc or lock her down.

That'll be all, I hope people can try out this build and discuss their thoughts and ideas on it!