I've been playing league since S6 (although I took multiple breaks), and started playing much more during Covid. I played on NA with 200ping, had enough, and changed to JP server, where I hit challenger within three months playing Aatrox. I peaked 1,091 LP and ended Rank 9 in S11 under the name "COJA".

Aatrox is not a traditional Fighter. You must space very well, moving accordingly to your opponent, almost like kiting. With 4 Skillshots in his kit, all of varying size and purpose, If it werent for his melee autoattacks, Aatrox would fall closer to a mage than a fighter. Although Aatrox heals for all damage dealt including autoattacks, you should ABSOLUTELY NOT PLAY AATROX LIKE AN AUTOATTACKER. Unless you are very ahead, Aatrox must fight smart. Into a number of matchups, you should poke majorly with Q, going in for extended trades only if the situation calls for it.

Experience is INCREDIBLY important on Aatrox. Not only does it make you tankier and deal more damage in every way, having a level gap on an enemy makes it doubly as easy for you to fight them. Of course you're stronger, but them being weaker means that they might not even be able to damage you. Aatrox is very good at fighting weak opponents, which is why setting your opponents behind is very important. This is why I believe wave management is crucial.

You shouldn't blindly push the wave. Here's just two examples of ways to use wave management to debilitate your opponent:

Slowpush multiple waves into a turret while poking the enemy whenever they try to farm, then crashing it to have the wave bounce back while poking them off of your caster minions.

Freeze a wave to force the enemy to overextend and allow you to zone them off exp and gold (or even run them down if you get a favorable trade), or just Freeze to be in a safe spot and get a bit of health from damaging the enemy when they try to break your Freeze.

I see people misuse Aatrox ultimate by failing to get the most use out of it.
In lane, Aatrox ultimate should be used when you either know the enemy is committing and you need your ultimate, or directly AFTER using Q1, BEFORE using your Passive auto, or AFTER landing your W pull. These are the biggest spikes in Aatrox's damage and you want to be able to use your ultimate to buff the maximum amount of damage possible. Aatrox ultimate may be cast in teamfights when you not only think but KNOW you can kill an enemy to get the reset. (If you see a low health target fleeing and ult to use the movement speed, this is a good use). (If you ult to engage into a group of 3 this is not a good use)

As Aatrox, you should be difficult to engage on, and once your opponent gets impatient and tries to engage on you, you should punish them badly.

Aatrox is built on dealing damage, so items like Seryldas are very important. Also, because the majority of Aatrox's healing is Flat, you generally prefer Resistances over flat Health. The three favorite stats of Aatrox are AD, AH, and Resistances.

A tip that can be applied to most characters is if the enemy has high Armor/Magic Pen, to build to counter it. If you have 40 Magic Resist and the enemy has 40 Magic Pen, building a Null Magic Mantle means going from 0 > 25 Magic Resist, (As effective as getting 25% more health)

For Starting Items, DBlade is better if you are going to be taking no damage at all, or when you do take damage, it will be from an extended fight where you need the extra 8 AD instead of the anti-poke healing from DShield. Think of it like Bone Plating and Second Wind. However, into basically any tank or non-diver, you want DShield. If you're unsure, take DShield. DBlade is a riskier item which can potentially win you a fight, but is less reliable.

Because of how Aatrox works, the best item on first back will be either Tabis (to not die whenever you try to fight), or Ironspike Whip (to make the enemy die whenever you try to fight). Phage is generally inferior to both, unless you are playing vs a tank which has gotten out of hand. Refillable potion is also one of the strongest early laning tools in the game. Cloth armor + Boots + Refillable can and absolutely will save your lane.

These are the highest tier items for general gameplay on Aatrox.
Goredrinker + Steraks work fantastically together, giving you additional means to sustain and extreme teamfight durability. Deaths Dance gives three of Aatrox's favorite stats. AD, AH, and Resistances. The "Ignore Pain" passive allows you a bit longer to complete your Q animation, and can nearly triple your healing on takedown. (These three items also have innate healing which is further increased by your ultimate)

Serylda's Grudge works a bit differently. Instead of leaning towards the "Unkillable" side of Aatrox, it grants massive kill pressure on any target. The flat offensive stats it grants work exceptionally well, and it works much better than Black Cleaver, even when a bit pricier and theoretically worse, because of the instant 30% Armor Penetration it grants, and its "Bitter Cold" passive. Even on your 0.25s knockup, it still slows for another 0.75 seconds afterward. It's even a tiny upgrade to your W slow (25% > 30% for one second), and also is applied on your W pull as well, effectively increasing your W slow duration from 1.5 to 2.5 total seconds if the pull is completed.

Boots are self explanatory, but I will explain anyway. Plated Steelcaps are almost a necessity against many matchups (Wukong, Sett, Irelia), granting much needed durability and movement speed so you can reposition and/or run away. Mercury treads generally are not needed as you will be taking tenacity runes anyway, but should still be bought into a high CC enemy team.

All of the following items also work excellently. I cannot really recommend any item outside these except for maybe extra resistances vs a high AP/AD enemy composition.

Silvermere Dawn is often overlooked, but I think it is a pretty good item on Aatrox. Since you must continously be dealing damage to stay alive, the cleanse and temporary tenacity buff can win you a teamfight singlehandedly. The flat stats it grants are also pretty useful. Very good into semi-unavoidable CC.

Guardian Angel, very good for front-to-back fighting. Your ultimate also increases the healing granted from revival, and can reset multiple times during your stasis. Stopwatch is also a handy component.

Chempunk Chainsword (or at least Executioner's) is going to be necessary for side laning vs Divine Sunderer or champions with innately high healing. I cannot recommend Bramble Vest after the nerfs.

Black Cleaver should only be built if you don't need the slow and instant armor pen from Seryldas, or if you already have Seryldas but still need more damage. There is nothing wrong with this item, it provides a lot of what Aatrox needs, but I think there are several other items which must almost always be purchased first.

Anathemas grants damage reduction vs a single target. You should never buy this item so that you can side lane vs someone. Only buy this item if one person on the enemy team is extremely strong and you need to be able to survive their burst.

Spirit Visage works well with Goredrinker/Sterkaks/Deaths Dance, amplifying their healing and shielding, as well as your own innate healing. Although it does not provide the most magic resist, its passive can make you far tankier than any magic resist item in the right situation.