Those are the basic Summoner Spells. You can go for Teleport too, since Aatrox's potential gank on bot lane is heavy because of his engage due to his Q and his pretty decent slow from his E.

Starting with Doran's Blade. I believe it is better than the Cull, since it gives you better sustain with the health stat and also with the masteries you go to a nice 5% Life Steal.

Boots. I suggest Mercury Threads because Crowd Control abilities stomp Aatrox, so you need the Tenacity.

First item, for the most cases, is Blade of the Ruined King. You need this Attack Speed and Life Steal stats it gives, it's passive is nerfed a little bit but now the extra damage applies Life Steal too, which is nice. And don't forget about the active.

Second item, depends on what enemy you have and how well you have performed on your lane. For example you can either go for Death's Dance if you dominate the lane. If you have a decent lane, no kills just farm and the opponent, due to his champion's kit, can win the lane, go for Spirit Visage. Another choice, when facing an AP champion is Maw of Malmortius, which gives you extra sustain, a lot of damage, AND BOY THOSE PASSIVES, or you can go with something that is suitable to almost every situation, Sterak's Gage which gives you extra bigger range (size it is this time), extra damage and the shield that helps you to sustain a lot more with the damage being taken to the shield and help you activate your W's passive a lot more times. Then you complete the item which suit you the most.

Remember of Randuin's Omen, which is a pretty nice item if you split push a lot, since you shrink the enemy's damage output because of that Attack Speed and Critical Damage Reduction.

Guardian Angel is a must-have item for Aatrox, because you revive an extra time, and with the Blood Well untouched! That means you still have the Attack Speed from the passive even if you go down, and given the time to ressurrect you, you might get some Cooldowns from the Skills you need at that time being.

Here is some of my tips.

[Passive] - Blood Well: It is pretty clear. Fill that **** up for more DPS and more Health from the revive!

[Q] - Dark Flight: Gets 10% of your current HP, so think more before you use it while in full life. The knockup is not that hard to land, though it is quite dodgeable too. Prefer not to engage with Q if you are not sure that you'll kill the target, since it cost a lot of HP and you'll lose your disengage. Better to knockup a slowed target (which is very easy), or when you are in melee fight, cause it takes less to time to land. It is also a skill that will help you full the Blood Well a lot faster. Also if you use it, while you are half HP or so, you might benefit from W's passive if used correctly.

[W] - Blood Thirst/Blood Price: Gives you excellent sustain/damage. You can use the third hit to poke the enemy with Blood Price (kinda the logic that Jax uses), and the hit Blood Thirst to get HP back. Be careful when you are low because that HP that you lose might save your beautiful butt.

[E] - Blades of Torment: This is your main poke skill. A bit hard to land when the enemy is a bit far but does a pretty decent amount of damage and the slow duration is SUPERB! A nice mass skill that can be use for wave clear too, and can help the jungler more because of that slow duration. It uses little of your life so it's definetely effective.

[R] - Massacre: This AOE skill is awesome! (Tip: You can R+Flash into the direction you want, same as Lee Sin's ulti logic) This freaking buff is awesome. Gives you range, attack speed, does a lot of damage and one of the most awesome part is that fills your Blood Well a percentage (20%), depending on how many Champions it hits. That being said, if you're about to lose your Blood Well, you can keep to activate it AFTER you get revived to gain some more attack speed after the Blood Well being consumed. Or you can activate it while you're being tower dived, to fill more your Blood Well and get more chances to survive respetively.

Ferocity

>Fury: Pretty decent stat and matches perfectly for what Aatrox needs.
>Double-Edged Sword: We have enough sustain already from our W and we have to increase the somewhat damage we're losing from not going 18 pts on Ferocity.
>Vampirism: Gives Aatrox one of his basic needs.
>Oppressor: With 1.75-2.75s (which is one of the biggest timers for slow currently in the game) slow you get a lot of extra damage AND from the start of the game.

Cunning

>Savagery: Better lasthits, safer farm.
>Assassin: The extra damage is VERY nice, since you got 12 pts on Ferocity and this damage kinda makes up for it.
>Merciless: More extra damage, I like it.
>Dangerous Game: It is hell better than bandit.
>Precision: Armor penetration is what an AD champion needs, and Aatrox also benefits early from Magic penetration since your E is your main poking ability and does magic damage.
>Windspeaker's Blessing: The top-tier Keystone for Aatrox in my opinion. You gain half a Spirit Visage for free! The reason we pick this I think it's obvious. You know. Even more sustain!

Well, pretty basic runes. Depending on AD or AP lane you get either the flat Magic Resist runes or the scaling ones. I think there is no need to analyze the runes, they are pretty clear.