Hello! This is my guide for AP Aatrox. I must admit he's pretty fun to play but this build is absolutely serious as it actually works really well. If you don't believe you can see some of my match history.
Overall he's very good for bursting down squishy enemies, but unfortunately doesn't do so well against tanks.
His power spike is Hextech Gunblade and acquiring Massacre from level 6.
With no doubt the longer the game, the stronger he becomes.

Start with Doran's Shield to heal up the health you use for your abilities and for possible enemy poke, because as you're playing AP, Blood Thirst doesn't work as well as it's supposed to.
With first recall buy Hextech Revolver (if you've got enough Gold obviously).
Try to rush Hextech Gunblade, healing passive make your abilities absolutely free instead of costing health, it also give both AD and AP and as we know both scale well with Aatrox's abilities. The most important thing though is Hextech Gunblade's active. It helps a lot in bursting down enemies in early and mid-game, with quite short cooldown.
Sorcerer's Shoes give you magic penetration, which is quite obviously needed if you really want to burst down your opponent.
Nashor's Tooth, along with very useful CDR and Ability Power gives you an amazing passive which works really well with Aatrox's attack speed from his Blood Well and Massacre.
Luden's Tempest and Rabadon's Deathcap are your main burst source, as they give you the most AP.
Last item depends on what enemies have and what do you need against them, you can obviously judge it yourself by just looking at enemy team.

Max Blades of Torment first to maximize your poking potential in lane, it's also your main source of damage except of Massacre. Add Blood Thirst early on in the game, to be able to heal up from enemy poke, as you're most likely to play against a ranged mage, who will poke you a lot. When you get level 4 add Dark Flight to be able to knock enemy up when your jungler ganks. Max Dark Flight second to lower its cooldown, as you need it to initiate the fight or escape.

For masteries I recommend either 18/0/12 (with Deathfire Touch ) or 0/18/12 (with Thunderlord's Decree ). I personally prefer Deathfire Touch , that's why I included it in masteries above, but Thunderlord's Decree works just as well on him.
With masteries you obviously go for everything that gives you Ability Power or just % of damage dealt.
On the right of your mastery tree, in Resolve column you should take everything with Health Regeneration to heal up the health you use for your abilities.

Take runes as shown at the top of the page. They give you:
- Magic Penetration to help you deal more damage and burst enemies.
- Cool Down Reduction to help your abilities refresh quicker, as AP Aatrox is, with no doubt, completely useless without them.
- A bit of Magic Resist, to equalise it with your Armor, as you get more base Armor than Magic Resist.
- Quite a lot of Health Regeneration to again, heal up for your abilities and the damage you take from enemy poke, as you're most likely to play against a ranged mage, who will obviously try to poke you down.

Basic and the best possible combo on AP Aatrox is:
- Hextech Gunblade, to deal some damage and slow down the enemy.
- Blades of Torment (E), to keep the slow going. It's also much easier to actually land it on the enemy with the previous slow.
- Dark Flight (Q), to knock enemy up.
- Massacre (R), to deal the most damage and finish off the enemy.
- IF this was done in teamfight, quite smart thing to do would be Flashing out from between the rest of the enemies, unless you trust your teammates enough to know they won't let you die there.

First off,

Remember to take your correct Runes and Masteries, as it's really hard to survive early game with anything different. When you start the game, probably even on the first levels you will see how useful all this HP Regen actually is on our AP Aatrox.

In Early Game

Tell your jungler NOT TO GANK until you acquire level 4 and therefore Dark Flight. With that you can knock enemy up and get easy kills with a good gank. Overall, try to play him deffensive, only farming and poking enemy with your Blades of Torment. If enemy pokes you down too much, just back off for a moment and your incredible [/color=green]Health Regen[/color] (with everything chosen correctly you get almost 6 HP PER SECOND AT LEVEL 1)

Mid Game...

When you get level 6 (and therefore Massacre) AND your Hextech Gunblade you can start playing much more aggressive, BUT only if you've got BOTH Massacre and Hextech Gunblade' active ready.

Later...

As you get more and more items and levels you only become more powerful, so with every another minute you can play more and more aggressive, up to the point where you're able to easily one-shot enemy Carrys.

In Teamfights

Try to ignore the enemy frontlane and jump directly onto enemy damage-dealers. When the teamfight phase starts you should easily be able to one-shot the carrys.