As someone who has played a lot of the current Aatrox, playing this new one feels odd, but not in a bad way. Somehow even though everything is different he still feels Aatrox-y. Somehow. Keep in mind this is gonna be about Top Lane Aatrox. I haven't had a chance to try him in the jungle.
As for what I've actually been doing, I've been using this setup: Conqueror, Triumph, Legend: Tenacity, and Last Stand for my first tree and then going Nimbus Cloak and Transcendence for my secondary. My build has been Cleaver into Titanic into straight tank so far, I'll probably try other things as I get the opportunity.
Here's my thought process. Conqueror seems good to me because it gives a lot of damage without having to invest in damage items, helps out his powerful early trades, and keeps tanks from walking all over you. Basically the same reasons it is good on live Aatrox. Triumph is just the usual standby, however presence of mind might not be a bad pick, given the ult cdr on it. Tenacity is good to keep you from getting kited and general anti-cc. Last Stand because your ult revive lands you at the perfect HP to use it. As for Nimbus Cloak, when you combine that with your ult MS, you suddenly move faster than a speeding bullet. Great for keeping people in range, or bolting at them to start a fight. Finally we have transcendence for the CDR. More CDR means more doing what Aatrox does.
As far as my build, Going cleaver first is obvious for its synergy with Aatrox: cdr, hp, ms, armor shred, and a little ad. Then follow up with a titanic, making you able to waveclear without having to spam out Q and giving you a goodly amount of damage and dueling potential. After that you go full tank, making you very disruptive in a teamfight. Going visage if your opponents are AP is of course, going to be wonderful with his healing passive on E and going Randuins if they're AD adds even more CC to your disruptive kit.
In my opinion, the new Aatrox is going to be someone who runs at the enemy team with ult to get a fight started and remains in the fight as a source of disruption and mild to moderate damage. Given his healing is now on all the physical damage he deals, I imagine he can stick around for quite a while in a teamfight.
For my final thoughts, I'll just put a few things I'd like to see changed.
1.) FIX YOUR TOOLTIPS! I WANT NUMBERS, GAD DANGIT! But seriously, I had to look it up on Surrender@20 to know how much AD his ult gives. That's completely unacceptable. Nor does Q tell us how much each one increases in damage. It's lazy and sloppy.
2.) I'd like it if his Q felt a little more rewarding to land the blade with. maybe up the knockups by .25 secs or else put a big knockup on the 3rd Q.
3.) Something needs to be done about his passive. Either make it so it only procs on champions or reduce the CD, because it feels really bad to have to wait for both your passive to be up and an opportunity to start a fight to happen at the same time.
4.) I think the slow on W needs to be upped, at least at rank 1. Given how much mobility is to be had in league, it's pretty easy to ignore a measly 15% slow. Especially as W seems to have emerged as Aatrox's one point wonder.
5.) Finally Aatrox's ult needs something to give it a bit more oomph. The MS is nice, the minion fear is pretty inconsequential, the bonus AD is meh, and the revive is good. For an ability called "World Ender" it's pretty limp. Maybe give it a short fear on champions? Maybe have it do damage on activation? Maybe when your ult is active, the bonus AD you get from E is extended? Maybe have it give your passive lower cd/ 100% uptime for the duration of ult? I particularly like that last idea, as it gives Aatrox more attack range, making it somewhat more difficult to escape and giving a little of what people like about his current ult.
If anyone wants to give any thoughts of their own, I'm all ears.