You are not really AP champion, but you can make it work with two specific items!
You rush Hextech Rocketbelt #1, Liandry's Torment #2.
If you are against many HP stacking champions, you can rush Liandry's Torment first.
You get Cosmic Insight rune to use Hextech Rocketbelt more often.
You poke with Hextech Rocketbelt and First Strike to earn gold and burn enemies.
After core items, you adapt. You can play as a tank or as a damage dealer.
How to play
General
Use Hextech Rocketbelt active to poke your enemies all the time, earning gold with First Strike.
Adapt to the enemy team. Build tanky or more damage. You can play as a utility tank, poke, or even you can go some AD items to still have plenty of damage. Depends on the team and how you want to play.
For utility tank, remember that Liandry's Torment works good with Serpent's Fang (to counter shields) and Oblivion Orb (to counter healing).
Items
If you are doing really well and want to have even more AP dmg, you can go Lich Bane, Cosmic Drive and The Collector (it procs and works nicely with DoTs).
If your team needs strong Physical damage, you can just go AD items.
Cosmic Insight reduces the cooldown of all active items, so Zhonya's Hourglass really works well with this build.
Runes
Wait for the First Strike and proc it with your combo, to deal damage and earn gold.
Be careful to not get poked, because you waste First Strike procs.
Your main combo is First Strike + Hextech Rocketbelt + W + Q + E + go in.
Your alternate combo just for poke is First Strike + Hextech Rocketbelt + W + Q + E.
Tips and Tricks
Riot
Saving energy and spell cooldowns until you use your ultimate will maximize Death Mark's damage.
Quickly double tapping Living Shadow will teleport Zed to his shadow immediately for rapid escapes.
Placing a Living Shadow before using Death Mark can give Zed a safe escape from combat.