Hey guys, I'm going to keep this short and simple. I play Lux as a AP carry. I will give you the core items of the build and the rest is up to you on how to spice things up.

Mark
Magic Pent, thats all you need

Seal
Mp regen flat or per/lvl either one is fine (With Doran's Ring you will be regening really fast)

Glyphs
Cooldown per/lvl or flat whichever you perfer
Flat Ap is also viable here too

Pros
Burst enemies like a boss
Rape people long range
Has a cool smurf, up to 2 smurfs at once
has a good aoe slow and damage

Cons
Squishy without giving up some AP
Requires some skill with skillshots
CD could be ridiculous at early levels

Same as most spellcasters should get.

I start with Doran's Ring and then go for boots (Magic Pent. Boots, CD boots, or Merc treads). After that i get another Doran's ring and if im not getting more cs then the enemy champion or not laning/soloing well i would go for another doran's ring. The 3 Doran's Ring will give you insane MP regen, Which allowing you to farm/lane more. This also lessen the chance of you being in a fight without mana which makes you completely useless in the fight. As a spellcaster you need to always have mana during mid game to end game. Everything else is however you want to build her. Usually I would go for deathcap and void staff (if they get MR) if not, i would go for some survival such as banshee

Max Q or E first
Somewhere from lv 7-18 get a W but don't max until forced to

I Was Only Pretending ^^
During laning/soloing phase when your harassing or trying to zone out the enemy champions, They will be anxious to get my into their "zone". What you can do is pretend to run away and as you do, The enemy champion will then move forward in hoping they will go back into their "zone". What you do now is move forward towards the enemy champion and Q when u are certain or at least 90% sure you will hit him. Most likely you will be and you will be able to do the Q-E Combo or E-Q-R Finisher.

Oooooo Theres goes your HP ^^
Basicly you Hit the enemy champion with Q or E and then autoattack then Try to hit them with your other skill and autoattack them again. This allows for maximum damage and can usually get them quite low when uses 2-3 times. Although this could be dangerous as it takes a bit of time to do it so the key is time it right.

Yay!! I LUV GUESSING!! :DD
If you see them low, ULT! With lv 3 ult at 40% CD Reduction its a 20 second cooldown. See tryn get away with his E during a teamfight? ULT! Basicly its a guessing game, Good luck ^^