Level 1: Mainly look to CS and get to level 2. Aatrox is relatively weak at level 1, so don't try to force a fight unless you have your jungler. In addition, watch out for CC. It is easy for Aatrox to lose a lot of health early to a champ with some for of CC. You can look to poke enemies that are going for CS with your The Darkin Blad, but otherwise it is recommended that you do not interact with your opponent too much.

Level 2: At this point, you want keep CS-ing and look to get to your level three power spike. You sould still poke with your The Darkin Blade, and not look to go for an all in yet. But now since you have your Umbral Dash you will be able to poke your opponents better by using the following combo:
Q-(Q-E)
This short combo will allow you to hit two "sweet spots" on your The Darkin Blade, causing you to heal and deal lots of damage to your opponent.

Level 3: Once you have 1 point in each of your abilities, you can perform a high damage combo with Aatrox. The combo goes as follows:
Q-W-(Q-E)-Q-AA
Not only is this combo good for damage, it also heals you for a lot. You want to perform this combo on out of position champions. Since this combo allows you to hit all of you "sweet spots" on your The Darkin Blade, the opponent will not be able to deal much damage to you because of all of the CC that you are applying to them. 2 full combos is enough to kill your opponent, so choose your right time to strike.

Level 4-5: You basically just want to poke your opponent whenever you can and try to kill them with your combos. If you are playing against a more difficult matchup you just want to focus on getting all the CS you can. You might find yourself backing at this time to get a Pickaxe or even an Ironspike Whip if you killed your opponent.

Level 6: At this point, you can probably 1v2 ganks with your ultimate (especially if you have Ironspike Whip). The reason you want Ironspike Whip is because it will not only help you CS, but it will also help you stack up your Conquerer for more healing.

Mid-game: During the mid-game you generally want to group with your team and get objective or picks on your opponents.

Team Fights: During team fights, as Aatrox, you usually want to focus the ADC. You want to do this because at this point in the game you can one-shot the with one full combo, allowing you to get an easy reset on your ultimate and eliminate a lot of damage on your opponents side, which will most likely allow you to win the fight.

Late Game: Aatrox tends to fall off late game because your opponents tend to by Grevious Wounds. It's not that Aatrox doesn't scale very well, it's that a lot of champions outscale him and as an AD spellcaster that relies on his abilities to heal, if you don't hit your abilities you will be out-matched by your opponents.

If you read all this way, thank you so much! I've worked really hard on making this guide and I hope that it is helpful!