• Rememeber your passive: Illumination, it's simply OP. Every time an enemy is marked by one of your abilities try to make a basic attack on him to trigger it (in early game the damage can be surprising). And take profit of it for farming.
• Keep in mind that your Final Spark triggers and refreshes Illumination, so you don't need the basic attack before the ult to maximize the damage
• Use your Lucent Singularity to continuously poke your enemy and farm.
• A very powerful combo for poking is Light Binding (Q) + basic attack + Lucent Singularity (E) + Lucent Singularity (E) (to trigger it) + basic attack
• Once your opponent is at 2/3 or 1/2 of his maximum health use this combo to kill him: Light Binding (Q) + Deathfire Grasp + basic attack + Lucent Singularity (E) + Lucent Singularity (E) (to trigger it) + Final Spark (R) + basic attack (if he is still alive), feel free to start with Deathfire Grasp if you are sure to snare your opponent with Light Binding right afterwards.
• If an enemy is very good at dodging your Light Binding, try to send him your E ( Lucent Singularity) without triggering it in order to lure him into your Q ( Light Binding), so once your enemy is snared you can continue with one of these combos:
  • basic attack + Lucent Singularity (E) (to trigger it) + basic attack
  • (if you can kill with this) Deathfire Grasp + basic attack + Lucent Singularity (E) (to trigger it) + basic attack + Final Spark (R) + basic attack
• When you almost killed someone, but he has to much health to be killed by a basic attack and all your "hurting" abilities are on cooldown, IF you have Lich Bane you can make the very powerful combo: Prismatic Barrier (W, the shield) + basic attack, so that your Lich Bane can kill the enemy
• While you make a combo your enemy will very likely try to hurt you, so add your shield ( Prismatic Barrier, the W) somewhere between the begin and the and of your combo
• When you are running for your life and some enemies are chasing you, DON'T FORGET that you can snare 2 enemies with your Q ( Light Binding) and slow them with your E ( Lucent Singularity).

Usually Deathfire Grasp is a situational item, and you buy situational items in the late game. But I think this is a bad habit if you play full AP, because of it's TOTALLY CRACKED active: I am not speaking about the part where it takes 15% of the enemy max health, but about the part where it says "Amplifies all magic damage they take by 20% for 4 seconds" that is very useful for each mage that deals loads of damage: IT IS FREE BONUS DAMAGE (once you bought the item).
And as you see in the section above I use it often in my combo. In early game it helps very much to dominate your lane. and in killing very often an enemy in 1vs1.

I don't like Athene's Unholy Grail because it doesn't give you enough Ability Power and after the rework that nerfed the mana regeneration i like it even less. Although the passive Mana Font (that increases mana regen by 1% for every 1% of max mana missing) is very useful in early game, but Athene's Unholy Grail costs 2600 of gold that is way to much for an item I only want in early. But good news!!!!! There is Chalice of Harmony that has also Mana Font as passive and costs only 880 that is way more affordable for early game. By the way it has 7 mana regen per seconds instead of the 10 of Athene's Unholy Grail. So I think it's better to take only the little version of Athene's Unholy Grail: Chalice of Harmony

Later in the game you should sell this item because you are not that mana hungry anymore.

I explained above in the section WHY Deathfire Grasp as first core item? the reason of Deathfire Grasp. Here is the reason for the rest.

• Lich Bane is a main Item for Lux, because of her passive Illumination: not only you deal tons of damage with your abilities, but you will make your enemy suffer even more than with only your passive with your next basic attack
• As we are playing full AP it's obvious to take Sorcerer's Shoes as movement item
• Rabadon's Deathcap and Zhonya's Hourglass are mage items that are mainly for Ability Power (and some armor is welcome with Zhonya's Hourglass).

For the last/situational item:

• If your ennemies are buying magic resist items (or they have above 60 of Magic Resist), then you should buy Void Staff
• If you think you need more health and still want a good deal of AP Rylai's Crystal Scepter is what you want
• If you want to last more time in the team fights, some cooldown reduction and loads of AP, then you should take Will of the Ancients. That will turn your tremendous damage in health for you
• If you think you need more Magic Resist then take Abyssal Mask (that will also reduce by 20 the Magic Resist of nearby foes)
• If you think you need more Armor, than simply take another Zhonya's Hourglass (the 120 AP are very welcome, unless there is a Veigar that wants to ult on you)

I always keep in mind that Light Binding scales 70% of AP and that Lucent Singularity scales only 60%, so the E ( Lucent Singularity) deals less damage than the Q ( Light Binding). But as Lux is a combo champion, it's better to keep these abilities close in level, they are always at 1 or 2 levels of distance.
Never forget that everything depends on the situation: if you keep failing your Light Binding simply invert the order, so that your occasional pokes with your E ( Lucent Singularity) deal more damage: