I start with Doran's ring and two potions all the time.
This allows my Q at level 5 to one-shot the ranged minions at max range Q.
I always get Dark Seal and Mejai's, because one kill, will boost your Q damage Significantly. She is a Snowballing champion, She exists to one shot.

Luden's first item is obvious, with the extra Q damage and Ludens finishing the Melee minions, and one-shotting ranged minions.

I take lich bane, because, it basically doubles / triples my passive damage. and also tower shred. people usually go for morellonomicon for the grievous wound and also the magic pen with health, but I don't think you need those to one-shot anyone when you got significant extra damage increase in a long-ranged ability that you can barely punish. the numbers say morellonomicon is better overall, but I still love the passive amplification damage.

The new item Horizon Focus is pretty good on zoe, it makes your q after your e deal insane amount of damage. you can also proc the item buy qqr combo without your e but its pretty wonky and unreliable.

Deathcap, I don't need to explain this.

Protobelt is a really good item for clearing waves, but mostly for one-shotting enemies from outside of their vision. you can do your usual QQR combo and press E and proto to deal extra 50% of your QQR combo. This item cannot be built with the ludens after season 11 which is sad, but horizon focus covers that empty spot.

This combo is essential and used for long-ranged one-shot utilizing her ult in her specific time during her second cast of her Q.

This combo used to be so ridiculous back when she was released. because she could abuse the combo to make her Q the first thing to appear instead of herself, but Riot thought it was way too broken and kind of patched it. It is still possible to execute, it is not a good as the old QQR or as easy it is.

How to execute QQR combo

1. Cast Q backwards in a way where it is not obstructed by anything.
2. Walk in the direction where your target is.
3. Cast Q as it is about to run out in a predicted direction where your target is going to be as you keep walking forward into the general direction of your Q.
4. Press ult before/as your Q passes by your Character model. (The old Zoe was able to keep walking and ult right before her Q is about to disappear, which gave the enemy no chance to react to the high-velocity nuke.)
5. You can cast your passive auto and e or e and protobelt.
6. Congrats! You just one shot someone.

This Zoe build is mostly for the Snowballing.

With the eyeball collection, Mejai's, Dark Harvest.

Also for this style of Zoe play, you are REQUIRED to have decent knowledge on positioning and have mastered Zoe's max range Qs and QQR combo.

Level 1, cs normally and last hit minions with AUTOS. if you have to use Q to get 2 or 3 minions, go for it. make sure your lane is pushed up. this creates opportunities for you to gank or follow up in JG fight later on.

Level 2, This is where the fun begins. CS normally, if you see an opening for 100% E hit, go for it.

Level 3, This is where you can make exceptional plays with the random drops.

level 5, you can two-shot ranged minions with Q first and get melee with 2 autos or Q. This will definitely push the wave to the enemy tower, make sure to ward up the river and the back red buff bush, and keep an eye out. try to punish them without of vision Q as much as you can.

make sure you use this vision trick, where you start your Q from point A and as you are approaching point B recast your Q towards the minions. also, unless they are 100% hit chance, you are aiming for minions and also hitting the enemy at the same time. Watch for your Q hitting the raptors.
Level 6, is where you can one-shot people out of nowhere. They will be peacefully farming and see their health bar go down to 40% because of one Q. You can one-shot ranged minions by maxing out the Q damage.

After pushing your lane, start roaming to other lanes. The enemy mid laner will be busy last hitting under tower or walking back to lane.

After the laning phase is over, this is where you shine. make sure you have your surroundings always controlled and warded, so you could react to it.
Walk to a lane or space where you are safe from behind and you are out of the enemy's vision. when they are stepping in a pattern or killing minions, that is where they are the most vulnerable, since they do not expect a Q to fly out of nowhere.

During a teamfight, you want to have a distance between your team and the enemy, where they think you are not present in the fight, or do not know where you are, and when you see an opening of them roaming around or in the range of your combo after checking your surrounding of enemies, kill them. If you dont manage to kill them, its ok, you probably did a sufficient amount of damage enough to make the person useless for the fight.

And protobelt for the extra range that they dont expect.

Zoe burns through mana extremely quickly early game. Make sure you are not blowing through Qs to micro trade, especially when this build does not go mana flowband. During the level 6+ Laning phase, you can also ask your jgler for blue buff which gives you infinite lane clear, pressure, and presence.

Do not be discouraged by your garbage cs or KDA, even if you are 110 cs at 24 min and 2/7/4. If you just have your first item and Mejais, and win one teamfight without dying, you will easily get your lead back fast.

Also, do NOT sell your mejais unless it's being replaced by your last item or defensive item. The item is too good for Zoe's playstyle, especially you really aren't seen by the enemy unless they are Rengar, Twisted Fate, or Nocturne.

When you are up against those ult dash champions, just build stopwatch, and take barrier.