Lux is a ranged AP carry/nuker with an insane spell combo that can pretty much kill any enemy champion pretty easily if you land your skillshot snare. An overall balanced champion because of the amount of skill it might take to play her even with such a high burst.

These are the classic AP caster runes that most casters should probably be using.

Greater Mark of Magic Penetration are the only marks worth using as an AP carry/nuke, as with your Sorcerer's Shoes, you would be doing true damage if they have not built magic resist yet because of your 25 magic penetration + 8.5 while most champions have a base 30 magic resist.
Greater Seal of Replenishment gives you some nice early game sustain and you'd never want to run out of mana during the middle of your spell combo, because that'd just be embarrassing, now wouldn't it ?
Greater Glyph of Scaling Ability Power are used to give you a nice amount of AP throughout the entire game for a short boost in damage because you want to kill enemies as fast as possible

I go for 21/0/9 mostly because I play very aggressively and I really like to pull off some kills.Offense
Summoner's Wrath for the extra AP from Ignite
Mental Force for some AP
Sorcery More skill usage = more damage
Arcane Knowledge Just do it .
Havoc More burst
Blast More extra AP
Archmage Even more AP
Executioner Kill low health enemies more easily

Utility
Summoner's Insight More Flash's
Expanded Mind More sustain
Meditation More mana
Runic Affinity Considering you should normally hold blue throughout, you would want to hold it as long as possible

Early Game
You want to start with Boots of Speed] and 3[[Health Potions for your early game laning

Boots Avoid early skill shots and run around to farm.

Health Potion Gives you more sustain health-wise because you have Greater Seal of Replenishment for mana.

Mid Game
At this point you should have Sorcerer's Shoes, 2 Doran's Rings, and a Rabadon's Deathcap

Sorcerer's Shoes The magic penetration you get from these boots in addition with your Greater Mark of Magic Penetration gives you enough magic penetration to do true damage if they have not yet built magic resistance yet to counter you.

Doran's Ring These things are extremely cost-efficient and gives you a nice advantage throughout your midgame laning phase.

Rabadon's Deathcap A must-have for almost all AP casters and gives you a huge AP spike, giving you lots of damage. Always. Always.

Late Game
At this point, you should probably have several kills and be involved in some team fights. The key to winning team fights is, obviously to support your teammates in the back as a caster, while also bursting the enemy down before their team could even have a chance to win the team fight. To do the burst that you need, you need the items that provide you with lots of AP while also staying alive with some durability.

Rod of Ages Now you be asking,"Why get Rod of Ages after laning phase? It's a wasted passive." I cannot tell you how many times I have been saved after leveling up during a teamfight just from the healing and how tanky you could get with this thing. Lux is naturally a squishy character and this item just breaks that rule.

Rylai's Crystal Scepter This slow stacks well with your Lucent Singularity , providing for extra slow and oddly enough, you get to slow with your Finales Funkeln . By pulling off your full spell combo, if they do not die, they will never escape you can probably Ignite to finish it off.

Void Staff Counters tanks such as Galio and those who are stacking magic resist with items like Force of Nature

I take Flash and Ignite like any classic champion that would kill.
Why?
Flash can save lives and secure kills
Ignite gives you early game damage and just annoys people with the damage over time because if they escape and die, it's just sad.. for them. More kills for you and with your Summoner's Wrath , you can have even more AP which gives you more damage~

Illumination A great passive to have, as it gives you even MORE damage which is what this guide focuses mostly on and it's just awesome when you do bonus damage throughout your spell combo

Prismatic Barrier A nice shield that I take early on at level 4 to avoid jungle ganks and steal assists during a gank without really doing much if you can't catch up. This skill is really good against AD carries because it just blocks damage on the first shield and when it comes back, you block more damage from another basic attack.

Lucent Singularity I max this skill first as it provides the most damage, other than your ultimate of course, while also giving you great zone control. The slow on this skill, if used correctly is actually really good to prevent enemies from chasing or escaping during a team fight or duel.

Finales Funkeln Your ultimate. Your face melter. Your kill securing move. Your rainbow. This skill is what makes Lux different from other champions. This skill is a high-ranged nuke with an extremely low cooldown at level 3. When about to recall, don't be afraid to use it before you leave because by the time you come back, the cooldown would probably be off. I see lots of Lux players that are really good but just suck at using this skill which kind of irritates me. It seems to me like most players don't pay attention to this skill's cooldown for its amazing damage.

Skill Usage
Your spell combo is QERE which pretty much kills everybody. Your Q just snares them keeping them in place for enough time to throw out your E to a location that would be their escape and also giving you enough time to effectively land your ultimate. If they end up running away to your E, just activate it and they're gone.

As Lux, you want to farm as much as possible in mid lane. Most new players do not understand the importance of last hitting and not pushing the lane. If you push the lane, there would be easy jungle ganks which are hard to avoid because of Lux's natural squishiness so you pretty much die. To farm, you want to run around in circles around where your caster minions are until an enemy minion is at low health when you want to basic attack it to conserve mana, get the money, and the EXP. If the enemy seems to farm more than you, call in your jungler.

Well, this is actually my first build so don't hate :P
This build is also not yet complete so I'm still working on it.