Aatrox Needs Love
TL;DR: Just buff his
stats and scaling, even just slightly. See how that works, and work it out from there.
I think everyone else has beat this horse to death: Aatrox needs a buff or rework of some sort. These are just my thoughts on the matter.
His Q is so easy to cc and stop him mid-air. Most other cc takes priority over it too. That needs to not happen. There needs to be a balance there. His ult is another problem. It's purely AP with magic damage. The issue isn't the damage it deals, or what it takes to scale it up, and the 1:1 ratio is really strong (super fun on URF).
The issue herein lies in that Q and W scale solely with AD. E is the hybrid skill. R is AP. This isn't healthy, in addition to the tank/fighter/assassin issues he already suffers from. Can't build him mage, can't be a proper tank and absolutely pales to juggernauts, and can't be a good duelist when he gets destroyed by the slightest cc. Forget about assassinations, because if you fail, there is nothing that will even remotely save you. He's been out long enough for most anyone at low level to time cc and burst to kill you the second you come out of stasis. He may not get played a lot, but it doesn't change the fact he's so easy to counterplay against.
If you're good with him, and the enemy is less than decent, he'll look like a monster. At equal skill levels, it's kind of a toss up, but 1v1 will always struggle (That doesn't mean you'll always die though). And if your foe is better than you, just being Aatrox is almost in and of itself a detriment.
Everyone wants to make changes and stuff. If I had my say, just increase the AD scaling on all his abilities by very minimal amounts. But I do mean everything. The baseline damage on his skills aren't that great, although they're not terrible. Compare it to a Riven. And instead of doing some crazy ult change, just give it an AD ratio too. That way you can actually just build full AD and still have your abilities respond well.
Buff his baseline stats and growth stats as well. Nothing crazy. Small baseline boosts and increases per level will help him stay more relevant against his lane for the most part. Finally, just give him so AS back to his passive. Not what it was before because then nothing would have changed too much, but don't leave it where it is.
Just implement these changes, even in a PBE, and see how he turns out. These aren't changes to the kit (save for AD to the ult). These are just all stat buffs. As he is, he may not fit in with what you picture, but that's a good thing for a champion like him; he already stands out lore-wise. Just give him a few small boosts to what he has, and let him go. If it's bad, tune it back a bit. If it makes him strong and healthy, then all is right and well in the world. If it's too much, just tune it back or work it out from there.
What I suggest, and ask for, aren't far fetched or hard. Most champs already have full AD scaling. Please just try this in even a closed environment, and see how he is. One of the coolest, most enigmatic champions just needs a little love. He doesn't need a full makeover, but a slight tune-up. Thank you.