This build guide for Aatrox is for Top Lane. He does the fair majority of his damage through basic attacks and I consider him to be a melee carry. He is very survivable due to his healing much like Xin Zhao or Irelia and he is very similar to these heroes in design.

Pros
Has quite high base statistics
Very high sustain with Blood Thirst
Has a gap closer with a knockup built in
Gets a ton of free attack speed from his passive and ultimate
Gets an extra chance at living through his passive every 225 seconds
Does tons of % max health damage, especially with Blade of the Ruined King

Cons
He has only one gap closer
Has relatively low crowd control
He is immensely useless without items
Has very low ranges and very little answer to ranged harassment
He is very reliant on attack speed, so is countered by attack speed slows like Warden's Mail

Farming will always be an important aspect of the game, unfortunately for Aatrox, it's immensely important since he doesn't have a very high base damage from his kit. His ultimate is a lot of free damage, but not enough to carry the game. You shouldn't be concerned, however if the game is going late with Aatrox as every point of AD you get scales about 110% more than any champion without any attack speed steroids. You also essentially get double bonuses from Lifesteal through AD due to Aatrox's Blood Thirst.

Make Informed Decisions: Knowing your target and item-checking
Constantly check everyone's items in the game so you know what you should itemize. It also impacts your decisions on who you should focus as players building no resists or health are easy targets to dispatch. Alternately, because you have a % Max Health nuke, you need to watch players MR compared to their health to know who you want to hit with Massacre every fight to have the maximum impact.

Make Informed Decisions: Protecting your carries or diving the enemy carries
In the situation that your carries have higher range or scale better, it's generally best to protect your carries over diving the enemy. Enable them to carry you by protecting them. However, if they're a Graves or something, he's generally impossible to peel for and doesn't scale better than most enemy carries, you'll probably find it better to dive onto enemy carries to try to kill them.

Split pushing
Aatrox has extremely high dueling potential with his kit and if he starts to snowball in lane no one on the enemy team would be able to duel him. This makes Aatrox a great choice to be able to split push and with Dark Flight he will be able to escape from most dangerous situations

Team Fighting
Aatrox can server as a very tanky initiator with Dark Flight and Massacre spiking the enemy team out of a lot of health. However, due to Dark Flight's relatively short range, it's not an amazing initiate. The importance of knowing when and how to initiate is crucial to knowing how to team fight with heroes that aren't exactly geared towards it. You have to be mindful of your allies' and enemies' positions and make snap decisions. If you see an enemy on the opposite side of the map split pushing? That is your signal to go if your team is grouped. Allowing people to split push unpunished is pretty bad because they're getting XP and gold undivided while you sit diving XP 5 ways.

Be Mindful Of Blood Thirst and Blood Price
You can use Blood Price when you are at high Health and it will allow you to do significantly more damage as it makes your every third auto attack with the cost of your own Health. Remember to switch to Blood Thirst when your Health falls low 50% as it doubles the healing effect and win you the trade.

3rd Hit Toggling
You can prime a 3rd hit of Blood Price or Blood Thirst and swap the toggle to make a choice on which you want: damage or sustain. Be very mindful of this and predict how much damage you'll take from enemies so you can maximize your survival time and your damage. In your mind you should get used to having to count 1...2...Blood Price/Blood Thirst... similar to how Jax's passive on Grand Master's Might works.

Mind Your Blood Well Cooldown and Maintain Blood Well
You generally always want to have Blood Well at its max so that you get the most attack speed and so that if all goes wrong and you die, you come back with the most potential health. It's also important to be sure that your Blood Well is even available to be used assuming you die. If you're not confident in your own play, just be passive while you don't have Blood Well available.

Chaining Crowd Control
When going on the enemy laner aggressively it's often the best to use your Dark Flight first then your Blade of Torment. Blade of Torment gets narrower as it travels so hitting a target from far way is a bit difficult, if you are able to land your knock up first your opponent will not be able to dodge it.

Priming Blood Well
Before entering the lane, if you are not leashing a buff or invading the enemy jungle, you can start with your E, and spam it for a bit before minions spawn to build up blood for extra attack speed thanks you Blood Well. While you may lose some health, it gives Aatrox free early stats that will just be healed back with Lifesteal quints or Blood Thirst.