I build standard ap runes with ap marks, glyphs and quintessences and mana reg seals.
My masteries are ap too, with 3 points in magic resist to survive, because i picked ap damage in my runes.
My skill priority is r->e->q->w, to give a high amount of damage and stack the soulstealer.
I start with boots, to dodge skills and have a bit sustain. I buy 1 or 2 dorans rings to spam skills, stack the boots to sorcerers, for the burst and buy mejais soulstealer, because lux is a very ranged champ who stays very defensive and survives most of the teamfights.
If there are not much teamfights, I buy the catalyst for more sustain on the lane, if not, i rush the deathcap to burst everyone down.
After this, i buy the rod of ages for a bit more tankyness.
If the enemies stack magic resist i buy void staff next, if not lich bane, because of her passive.
My last item is the hourglass, for more damage and survives in the teamfights.

Lux has a high range, so I stay defensive and keep the focus on farming. If you want to poke, always throw an auto hit to deal as much damage as possible.
If u know someone will come mid use your e to show that u saw him and he cant gank you!
U will need much blue buffs to spam ur skills.
But the very important thing with lux is to stay defensive, so care of ganks and do not push the lane too hard.

Now you should have farmed well and bought your soulstealer. In the early teamfights u have to stay behind your team and try to q the ad and ap carry. Dont waste your laser too early and wait for cc on the carries to burst them down and be sure to hit them. Keep atention on yourself and stack your mejais as good as possible. Make sure to not die.
U can use the e skill to get vision in the bushes, but make sure to not waste them.
Try to safe teammates with your shield!

Now you should be able to burst the ad carry down with a combo, so try to use your range to kill him in the jungle, but always make sure to not die and not use your stacks on the soulstealer.
If u had a nice mid game, u will have 20 stacks, so that u can use the couldown reduction and to poke enemies with your ultimate. Make sure that you have the job to kill the ad carry, so wait with your binding to get him.
Try to get vision on baron with e and steal it with your laser. You can also use it to kill low life targets in the jungle.
The most important item in the very late game is your hourglass.
Make sure to get the perfect moment to use it.
One situation could be that they try to burst u.
Use it and try to flash out, then use q to bind and burst someone down.
The second situation is when you have cooldowns on ur skills. Use it and you will be able to use skills again without getting any damage.
Lux' q has a nic range and you can safe your ad carry with it.
So always make sure that you know if u have to use it defensive if your ad carry gets low, or offensive if the enemy carry is hyper fed.

Lux is one of the best ap carries.
The main thing on her is to stay defensive and use your range.
Your q is the main skill to burst someone down, so if you make sure to hit it, the enemy will have to back off.
Keep in mind to farm with your e in early game if you have a blue or much mana regen.
And always try to get assists or kills for your soulstealer, no one will angry about ks with this item!