Old Aatrox's bread and butter was his W, a toggle-able third hit AA modifier that could either deal bonus damage or heal him. New Aatrox's passive _(as of a recent change)_ now does bonus damage and heals him, and its cooldown can be reduced by basic-attacking enemy champions.
This means that he's actually not THAT far off from being able to play similarly to his old self. He just needs an AS steroid in his kit somewhere, or more incentive to build AS (ala the recent Garen changes).
Here's a rough idea, with guesstimated numbers.
_**Q: The Darkin Blade**_
- Second and third swing damage amplification changed from 25%/50% to 20% (+1% per 50 Bonus AD) and 40% (+2% per 50 Bonus AD)
- Sweet spot damage amplification changed from 50% to 33.3%.
- Swing speed on all casts now scales slightly with Aatrox's bonus Attack Speed.
_**W: Infernal Chains**_
- Hitting an enemy champion with Infernal Chains now grants Aatrox 30/40/50/60/70% bonus Attack Speed for 4 seconds.
_**E: Umbral Dash**_
- Passive healing % halved, now scales up to original value based on Aatrox's missing Health _(caps when Aatrox is at 40% Health or less)_.