WHY AP Ezreal?

His skills has pretty high AP damage, which is 2.65*AP in total. Katarina has 3.1*AP. Diana has 2.5*AP. Anne has 2.35*AP. Kassadin has 2.55*AP. Ahri has 2.86*AP. It's a rough reference but still enough to pursue you AP build is a practical scheme.

With Lich Bane and Spellsword , Mystic Shot will have 1*AP damage in total. Essence Flux has 0.8*AP damage. Arcane Shift has 0.75*AP damage. Trueshot Barrage has maximum 0.9*AP damage. These really make EZ a strong AP carry.

Ignite deals some nice damage. If you determined to kill, this would be best choice.

If you want to assist your teammate or pick a free kill, bring Teleport.

If your opponent has deadly killing ability in early game, Exhaust will be your savior.

Maximizing Mystic Shot is a top priority because it reduces all your CD at every hit.

Essence Flux is secondary to Arcane Shift because it doesn't reduce CD at higher level, but if you think you are pretty save in the fight, you can maximize Essence Flux after Mystic Shot.

I pick up Essence Flux at level 3 because it could harass my opponent really well since it won't be stopped by minions.

Mystic Shot is a significant part all the game, so I pick up greater mark of hybrid penetration.

Greater Seal of Armor are my preference, but Greater Seal of Health, Greater Seal of Scaling Health, and Greater Seal of Mana Regeneration are also nice choices. Make your decision from your opponent.

STARTING ITEMS
You are nothing before you get Lich Bane. These will help you keep farming in early game.

Tear of the Goddess
The tear is really a burden in early game since it doesn't have AP or AD, but both Lich Bane and Iceborn Gauntlet give lots of mana (750 in total). I think the passive of Seraph's Embrace will make these mana more beneficial. The tear can also relieve your mana hunger. You can directly remove the tear and staff from the build for early fight dominance.
Remember to use Mystic Shot and Essence Flux frequently when you're in safety. It will speed up the stacking of the tear.

Lich Bane
I consider it as first core item because it boost your Mystic Shot significantly by its unique passive and bonus movement speed is definitely important for laning.

Rabadon's Deathcap and Archangel's Staff
You can buy a Blasting Wand first. If your Tear of the Goddess is nearly full, then take Archangel's Staff, otherwise take Rabadon's Deathcap. Buy the other one afterward.

Zhonya's Hourglass
Your damage is pretty strong after you get Lich Bane, Seraph's Embrace, and Rabadon's Deathcap. This item can give you chance to survive in team fight. Dead men do nothing.

Iceborn Gauntlet
Really great item for EZ. CDR, armor, and slowing effect are definitely useful, but its lack of AP may not benefit you well if you purchase it earlier. Its slowing effect can live with bonus damage of Lich Bane, by the way.

Hextech Gunblade
A great item for 1-on-1 dueling. The slowing effect can make you easier to chase or kite. Both life steal and spell vamp it provides are useful for dueling. Its only disadvantage is the price.