Aatrox is classified as a Fighter and Tank. His abilities primarily scale over Attack Damage and boost his Auto Attacks. This makes him a strong dueler, capable of fighting other Champions one on one/many. As such he is primarily played in the top lane, but is also capable of being played in the middle lane or jungle.
Whether playing in lane or in the jungle, it is recommended that you level your Blades of Torment first as the increased slow time as well as the powerful, ranged, magic damage offers more per skill point than any of Aatrox's other skills.
It may seem appealing to maximize Blood Thirst first for additional sustain, however the amount healed only goes up by 10 base with each skill point (roughly 3.3 health per attack, lower than Doran's Blade)
Switch to Blood Price for early game harass against other melee champions. On the other hand, switching to Blood Thirst and getting one last heal can mean the difference between life and death when running from foes.
Dark Flight can be used as both an initiation move and a disengage, but be aware that Aatrox is still vulnerable to crowd controls, such as stuns and knockups.
Dark Flight has a very fast movement time and can be used to quickly jump over smaller walls to escape pursuers or chase down enemies who have recently used Flash or other blinks such as Arcane Shift
Using Blades of Torment first can allow you to get in a few hits and hopefully force an enemy to attempt an escape, after which you can follow up with Dark Flight for further punishment. Reversing the skill order will achieve the same effect but not with the same efficiency - Dark Flight has a longer cooldown and is a "harder" form of crowd control, thus encouraging more efficient use.
Additionally, the slow from Blades of Torment makes landing dark flight's knock-up easier by reducing the enemy's ability to dodge it
When trying to disengage, it's good to use Blades of Torment first (since it has a longer range and a shorter cast animation) followed immediately by Dark Flight (since it has a shorter range and a longer cast animation).
With enough life steal, Aatrox can leave Blood Price on indefinitely, as the amount of damage he takes every third strike is negated by the life steal.
Make sure to utilize the procs of Trinity Force properly or the breadth of mediocre stats it provides will be less effective than an item more focused on a single stat.
Youmuu's Ghostblade is especially useful once you fall below half health, as the bonus attack speed combined with the tripled healing from Blood Thirst makes you very hard to kill, even without your revive passive.
For the same reason, Sword of the Divine can be used, although it is relatively cost inefficient.
Aatrox can use all 3 armor penetration items effectively (Last Whisper, The Black Cleaver and Youmuu's Ghostblade). The choice depends on whether or not you need armor penetration early, and also whether or not your enemies are focusing on armor.
If you predict that your enemies are likely to get high amounts of armor, it is best to get Last Whisper during mid-game, helping you transition into your late-game phase better due to the high armor penetration and low cost.
If you feel that you need armor penetration more early, or that you need cooldown reduction, getting a The Brutalizer is a good idea. It can later be upgraded into a The Black Cleaver, which offers high armor penetration and some useful health, or Youmuu's Ghostblade which offers a passive that synergizes fantastically with Aatrox, as well as some critical strike chance.
Randuin's Omen helps soak up damage in team fights and helps you survive long enough to take out high priority targets and the active slow helps catch running enemies.
Spirit Visage in conjunction with Ravenous Hydra results in fast healing due to the increased healing effects of Spirits Visage and the AoE life steal from Hydra. It also sync well with his Passive and Blood thirst as it also increases their healing.
Frozen Mallet is also a good choice since it provides 700 bonus health and an on-hit slow.
Guardian Angel is equivalent to a second Blood Well; they both revive Aatrox after death. This gives more survivability and allows for more opportunities to finish off targets.
When Aatrox takes fatal damage he is healed for the amount currently in his resource bar. Landing a fatal blow on him while the resource bar is low will result in him reviving with a lower amount of health.
Unlike other reviving champions, Aatrox'sBlood Well revive is rapid and grants him invulnerability during this time which allows him to quickly re-join a team fight after being dropped to critical health.
When fighting him, consider his resource bar as part of his own health. (For example, killing him with a full Blood Well will make him return with a large portion of his hitpoints and if the fight was closely matched, he will most likely kill you.)
Be aware that his passive (after reviving) doesn't provide any attackspeed anymore, because he consumes all the blood within it and be sure to look if he got the Blood Thirst/Blood Price buff while reviving.
When Aatrox casts Dark Flight towards your location, avoid being knocked up by moving away from the center of impact.
Most players will aim the ability right above you or right behind you. So, moving closer to Aatrox at the time he casts Dark Flight will likely make the skill miss you.
In lane, zoning Aatrox out of combat will quickly drain out his Blood Well making him a much more vulnerable target.
While he can be built as a fighter, Aatrox's stats and kit are similar to those of a melee ADC, like Fiora or Master Yi. These champions are devastating as long as they are able to attack, but are extremely vulnerable to crowd control, and while Aatrox's Blood Well helps him survive, he is no exception to the rule, as a well landed stun or snare can severely hinder his combat abilities.
In particular, not only his damage but his in-lane sustain are dependent on him auto-attacking. Champions with strong blinds like Teemo or Quinn will give him fits during laning.
Inflicting Grievous Wounds can reduce his healing from Blood Thirst.
However, Grievous Wounds does not affect the health he regains from his Aatrox'sBlood Well.
Without ability power (which likely won't be built), Massacre's initial damage is not a factor. The large attack speed and range bonus is a huge threat, but you don't have to worry about getting instantly burst-nuked by the activation.
The ability lasts long enough for most teamfights, but baiting Aatrox into using the ability to commit to killing you before a fight breaks out will remove the strong threat the ability presents in a teamfight.
However, it's recommended you only do this if you are a very hard to kill or catch champion, such as Lulu, Zac, Singed, or Karma.
Aatrox is auto attack reliant for not only his damage, but also his survivability. Reducing his attack speed by purchasing Randuin's Omen or Frozen Heart can help reduce both of those components greatly.