AP Zed is one of the weirdest ways to build the flashy Ninja of stealth.
If you buy the correct items, you will do tons of damage.

AP Zed Damage
226 dmg ludens
487 damage gunblade
505 damage lichbane
285 damage shuriken
220 damage shadow slash
163 + 50% of all magic and physical damage dealt to target = 862
total damage burst = 1723 no ult
total damage with ult is 2585 damage + 10% of targets max HP from Zed passive

Rush Cutlass. This will give you sustain and a slow. Play more conservatively than AD zed early as you don't do as much damage early.

Gunblade is a big powerspike and you can now kill people with a full combo + ult combo.

Buy ludens next as it gives insane damage and is easy to stack up. Lichbane is next for more burst.

At this point you should buy CDR boots to hit 20% CDR, 30% with blue buff.

Rabadons is built next for the massive AP increase to boost both Ludens and Lichbane damage.

Build Zonyas for more ap and defense.

Buy elixir of sorcery last as the true damage is just too good.

Pros:
does tons of mixed damage
people don't expect the massive burst AP Zed is able to do
good poke with ludens
doesn't need ult to 1 shot squishies

Cons:
Weak early game
No armor magic Pen apart from runes
bad vs tanks