i know that seastone trident can give you better farming, but when you skill up playful/trickster it get a shorter cd meaning that you can use it more often to avoid attacks or chase someone and when you have maxed it, you should be able to get the back creeps in 1 hit.
and its a must to take your urchin strike it gives you the best dps.

this build gives you the tankyness you need to keep yourself alive in team fights, and you get the burst damage you love,

Ryails crystal scepter: your seastone trident does a magic damage debuff which damage over time meaning that your ryails will work as a frozen mallet like your basic attacks will slow the target, and of course your other spells will too, and it gives a lot of hp, and as a melee mage you need that.

Lichbane: your spells got a short cd, and you will get insane burst when when using Urhin strike, and seastone trident, with your lichbane will burst them down like a boss.

Rod of ages: all of your spells have a large mana cost, and you will need that mana the Rod of ages gives you, and it also gives you a lot of hp which keeps you alive in teamfights, and alot of ap too.

Rabadons deathcap: Huge nuke ap dps enough said. :)

Sorcerer´s shoes: you need the magic penetration to also deal a good amount of dmg against tanks.

Will of the ancients: as a melee mage you need the spellwamp badly from Will of the ancients
and you will support your allies a little.

Flash: when you playful/trickster is on cd, you need a way to escape, and flash is good even after the nerf.

Ignite: it gives the last burst damage you need many times in early game.

I take the health stats and armor and magic resistants in the defense three course the amount of health you get now is insane and it will help you heavily in early game.

i take in some mana regen in the Utility three, be cause you need that heavily in early game to keep your mana bar up. And i take some spellwamp which helps you to keep yourself alive in a lane like top, and it stacks pretty well with your Will of the ancients. and i take some in the gold so you can quickly get your stuff and maybe buy a ward in start. summoner´s insight will help the cd on your flash.

i take the ability power in the offense three course he is an ap champion simple as that. i take summoner´s wrath to boost your ignite.