I take flash and ignite on AP Jayce because this will maximize your damage potential, and after all, you are the team's assassin.

Start the game with a Doran's Ring because this gives you the Ability Power and Mana to assassinate your opponent.

During the midgame you will start picking up massive damage items such as Deathfire Grasp, Rabadon's Deathcap, and Lich Bane. This increases your assassination potential.

The most important part about playing AP Jayce is understanding your combos.
To maximize your damage, here is what you do:
1. Switch to cannon form, wait several seconds, and attack your enemy once to reduce their magic resist.
2. Cast Hyper Charge, use Deathfire Grasp on your enemy, and immediately switch into hammer form.
3. Use To The Skies to jump to your enemy and slow them.
4. While jumping to your enemy, cast Lightning Field.
5. Since you have activated Hyper Charge, your next auto attack will be rapid, so your enemy has no chance to dodge the bonus damage from your first auto attack.
6. By the time Lightning Field has run out, your enemy should be below 20% HP. Finish them off with Thundering Blow, which does 20% Max HP at rank 5.
7. If the enemy is still alive, switch back to cannon form and shoot an accelerated Shock Blast to kill the enemy.

If you have correctly performed this combo, this deals 20% of the enemy's health plus base daamge plus at least 288% of your AP, thanks to Deathfire Grasp and Lich Bane. If you are skilled enough, you can get 4 Lich Bane procs in one duel. Assuming that you have 750 AP, this combo deals a minimum of 2160 Damage. This will be almost true damage because of your magic resist shred and because the enemy will likely be building armor because they think you are playing AD Jayce.

What are the advantages of playing AP Jayce over AD Jayce?

AP Jayce takes advantage of Jayce's 8 spells, which allows for insane Lich Bane procs. In addition, a lot of Jayce's damage is magic damage and he can shred magic resist with his cannon form auto attack. All in all, this build is great for killing the enemy before they can react, whereas AD Jayce relies on constant poke and generally can not assassinate a target.