Hello, this is my first ever guide, and I will be going over an IN-DEPTH look at Aatrox, and everything to know about him and his matchups and whatnot.
I apologize, but this is going to be a very long guide, so if your not interested in reading, I don't suggest this guide.
I'm just a guy who loves playing aatrox and wanted to share some insight about him to get more people to play him :D.
Enjoy!

You will always bring flash for aatrox, as having another way to reposition your q's is extremely valuable for him, especially your third q.
I always recommend tp for aatrox, as it allows for him to impact other areas of the map and get other lanes ahead since he does not scale very well in the late game.

You usually want to take dorans shield, as Aatrox is very weak early since his q crits don't do much damage, and you do not have the necessary cdr to combo smoothly as Aatrox.
It is also quite difficult to hit his q against mobile champs, so you will be even weaker since you cannot lifesteal as much from your damage, making you vulnerable to trades.
Another important point is Aatrox playstyle, which during laning phase, is a poke champion, so you will benefit from doran's shield the best due since you want long fights.
You will take doran's blade against immobile champions that cannot avoid your qs very well, or are comfortable with all ining as early as lvl 3.

Due to the fact that you most likely won't be able to rush Black Cleaver in your first few backs, sp you want to priotize cdr before getting black cleaver since aatrox is heavily ability dependent.
Getting warhammer and kindlegem is the cheapest way to get alot of cdr at the lowest price. When aatrox reaches 30% cdr, his q animations will become very smooth with his auto attacks and q auto q auto q auto becomes very viable since now there is barely any delay with your q cast, so reaching this point is Aatrox's early power spike. This is also why getting warhammer before Black Cleaver is important, as when you get hammer and kindle gem, then get your black cleaver, you will have exactly 30% cdr, since black cleaver gives 20% and warhammer gives 10%.
Warhammer also builds into your usual second item, death dance, as well, making buying these two items very optimal for aatrox. If you are even or winning, then you should opt to buy warhammer before kindlegem and vice versa. Getting phage is not very important for aatrox as cdr since again he is ability dependent and usually does not have a problem with chasing people down due to the bonus movement speed he gets from ult, his w slow and pull, and his q's providing cc. This is also why you cannot rush black cleaver first because rushing black cleaver would force you to get phage, which is not optimal for aatrox.

Due to the fact that Aatrox can lifesteal very well from his e passive, he is also made to be more squisher than other bruisers. His lifesteal is made to make up for the fact that he is squisher than normal, making his offence his defense.
Because of this, it is optimal for Aatrox to buy defensive boots like merc treads and tabis rather than more offensive boots like greaves and ionian. This helps resolves some of the squishiness he has in his stats.
Except for cc heavy teams, you should always opt for ninja tabis, as it gives more defensive stats than merc treads.

Black Cleaver and Deaths Dance are going to be your main Core items, although you may switch out Black Cleaver for ghostblade, if the team is squishy, or your against a ranged champion. A champion like aatrox, whose combos not super bursty, but are continuous combos over time, is the very definition of a drain tank.
This is why Black Cleaver and Deaths Dance are his core items, as Aatrox thrives on extended fights, since his combos are long, and he passivly lifesteals off of damage to champions. Black Cleaver works well, and Deaths Dance is one of the few attack items that increases aatrox lifesteal off of his q's. He is not an auto attack champion, so he cannot use items such as ravenous hunters lifesteal very effectively.
After finishing your Deaths Dance, this is one of your strongest power spikes in the game. This is because you will on average finish Black Cleaver and Death Dance at around lvl 13 as long as you are not very behind. At lvl 13, is when your passive healing from damage to champions has reached its max, and your q's, which is your main damage and trading tool, has reached its lowest cd and highest ad ratio. At this point, you do massive damage because of this and heal crazy amounts of hp, and is probably your peak strength in the game. This is why you should utilize this time to force sieges and objective control, since this is the point where you are strongest, and not very many champions should be able to contest you at all. At this point, even champions like fiora or irelia you can beat fairly easily at this point of the game.
Ghostblade is great against enemy ranged top laners as it helps you kill them quick due to aatrox high base ad and lets you catch up to them with its passive. Now if the enemy team also has a bunch of tanks, then don't opt for ghostblade even if the top laner is ranged.

These items work well against ad teams, since aatrox will thrive on fighting multiple people in team fights, these items should help to allow you to survive against the collective damage from the enemy team.
Due to the fact that you now get weaker after your Black Cleaver and Death Dance power spike, you must now opt for more defensive items, as you will act now more to peel for your team or as team disruption with your q, since you are very easily countered by armor items, and the enemy will have likely bought items with mobility or increased movement speed, effectively dodging your qs and limiting your lifesteal (thus your defense) and got grevious wounds as well. So you must now use your items to compensate for your squishyness rather than your innate lifesteal.
One very good item is steraks gage, as its passive gives you anti burst against the enemy team, and while you are still strong, this shields makes you an unkillable juggarnaut. So you will almost always build this item unless you need mr.

These two items ( don't have to build both) should suffice to provide all the mr you need, as spirit visage synergizes very well with your kit, as the bonus healing from spirit visage, acts to multiply the amount of healing with your passive healing from your passive and e, as well as multiply the healing increase when you ult.
Maw is an ok item as it provides more ad to your very high base ad damage, as well as its passive giving you a magic shield. You can build this, but it doesn't synergize with your kit as well as spirit visage. Because of this, I always recommend you build spirit visage first before deciding whether you should build maw of mortius.

These items are very situational and should be built on your first backs against certain matchups. For example, champions like fiora, lucian, vayne, and irelia are champions that have plenty of life steal and are auto attack dependent, so getting bramble vest before your cdr is recommended.
Hex drinker should be brought against ap burst champions such as teemo, since he can easily poke you out of lane, but its passive should allow you to survive dangerous situations where teemo can possibly kill you.
Executioners should be bought when you are against champions that life steal, but are not a terrible matchup for you such as nasus and vladimir.

This part is very standard and will not change for all the games you play as aatrox. You max q since this is aatrox's most important ability, it serves as his main trading tool and damage.
Then e over w since leveling up w only increases its damage and lowers its cdr, which is not very substantial. Your e will increase in healing % the more you level it up, so it is inherently more valuable, and cd goes down, which is your mobility and having a lower cd e allows you to reposition your q's more frequently, which is very valuable for aatrox since most of his sustain will come from his q crits.
You level up your ult whenver possible, as this makes your all ins incredibly strong and gives you massive stat boosts during sieges, or 2v2/3v3s.
Your passive also increases in its %hp damage as you level up, effectively allowing you to heal exponentially more since you heal 100% of the damage dealt to champions. This also serves to cause aatrox to become even more vunerable to armor stacking, as his q already are easily countered by armor stacking, causing less sustain, but now another intricate part of his kit, which serves as more sustain and damage becomes insignificant due to armor stacking.

In all your matchups, conquerer is the only viable primary rune that aatrox can reliably use, as his playstyle revolves around extended fights, only conquerer synergzies with this playstlye and no other.
Triumph is only rune that makes sense for aatrox to use as font of life does not do very much for aatrox since he is a bruiser, and presence of mind because he doesnt use mana or energy.
Because Aatrox's late game is terrible, you do not want to go blood line, as this only acts to make his late game not as terrible. Like applying air fresheners to a garbage bag full of garbage to try to make it more appealing. You usually want to go tenacity since you want to be attacking as much as you can in order to maximize his innate lifesteal. If the team you are up against does not have much cc, then you should opt for alacrity since his increases your dps and makes your split pushing better.
Next, just by numbers alone, last stand is the rune that aatrox can maximize the best from its damage from stats alone, so you should almost always go last stand, unless the enemy team has a bunch of tanks, then you should take cut down.
For the second tree, you will always opt for ravenous hunter since increased lifesteal synergizes extremely well in aatrox's kit. For the other, taste of blood should be taken if you need more sustain in lane, and are against a lane matchup, where you want to be poking the champion out the majority of the time and avoiding all ins until you are at a massive hp advantage. Examples would be like darius and riven.
Finally, since aatrox usually pokes, no matter the lane matchup, and is not very auto attack reliant and more ability based, double adaptive is best. And you can opt for mr or armor depending on the matchup.

In this variation, harder matchups like fiora and irelia, you should take resolve, as you won't be able to land your crits very reliably due to their mobility, so you should take bone plating against burst champions and second wind against poke champions. of course revitalize works best as it increases aatrox's healing, synergizing very well with aatrox lifesteal kit.

You should be generally playing safe, as your q do not have very much dmg from the low ad ratios. You should keep in mind to poke as much as you can with your q's during laning phase. At lvl 1, you should use your q to hit as many creeps as possible to hit a quick lvl 2 since your q is great for waveclear.
Once you hit level two, you now have access to much better trading,and should be using your e to either poke with first or second q usually, but never with third, as when you do, you do not have any more damage to return if the enemy fight you at that point. Due to your long q cd, you will not be able to use your main damage skill to trade back, and because you also used your e to do this, you don't have any escape tool either. Another issue is that your q is not very skilled up, so even if you manage to hit your third q, it will not do very much dmg.
Another tip, is how aatrox q cd work. Aatrox's q start as soon as you press q, but is reset if you press q again. So if you have used your first two q already, it is usually better in the early laning phase to hold off on using your third q, rather than resetting your q cd by using your third q since a, theres a very good chance they will be able to dodge your third q, and b, you will be able to use your first and second q much earlier, which not only are easier to hit, but hitting your first and second q will collectivly do more damage than the third q, effectively giving your more lifesteal. The only time you shouldnt follow this rule is if you are about to die, or are sure you can hit and kill the enemy with your third q. This is also applicable to the later stages of the game.

In his mid game, this is where his power spikes, as his q's do formidable dmg, enabling him to dish out massive damage and lifesteal. At the stage of the game, during sieges or team fights over objectives, your main goal is to dive the backline, as they are very easy targets for you to kill. The other beneficial part to this is the fact that because they are so squishy, you will be able to lifesteal a crazy amount of hp to allow you to survive the massive burst damage that the enemy team will do to you.
As aatrox, you almost never want to be the member that engages, as you are a squishy champion that is supposed to be tanky from your lifesteal. However, if you are engaging, you will most likely be targeted by the enemy team and cc, effectively cutting you off of your lifesteal. Without your lifesteal, you are just a very squishy champion, and will be killed extremely quickly. This fact in particular is what makes aatrox much more vunerable to cc than most other champions in the game.

Due to her insane amount of mobility, she has many way to dodge your q crits. So you should build bone plating and revitalize secondary due to her burst damage. You must play safe and dodge her e in order to win the lane. If you can dodge her e, then this is the opportunity to engage on her, as a portion of her damage is gone, and will be much more easier to land your q crits on her. Once she hits 6, you must play around her ult, as her ult allows her to easily dodge your q's effectiively if you were to go all in, she can avoid most of your damage with her q, while she is able to land all of her damage. Hexdrinker is also extremely valuable against her massive burst damage. Only use your w when you are sure she has used her last dash, as your w is easily negated by any of her dashes. Another trick you can use is when you get hit by her e, you can use your second q and dash backwards in advance right before she 2nd e's you. This is very hard to time, but if you manage to hit her in that time, you can easily win trade.

Although Aatrox does not scale well, he can definitely snowball hard. He is an extremely fun and skill expressive champion that makes him worth playing.

I will be updating continously, as there is still alot of information I still need to get into this guide, so expect more, such as updated early and mid game guides, along with late game and more detailed explanations for matchups and how you should approach them, along with what his role is and what he should be doing team fights. Although the types of champions that Aatrox synergizes extremely well with and the ways he can accomplish this in game. I do hope you enjoy the guide so far tho.