So I think I found a build on Aatrox that I like. If you think you are going to be team fighting early then get Guinsoo's Rageblade after your jungle item + Enchantment: Devourer. If you are still going to be ganking individual champions then get Death's Dance after Devourer. The thing about Guinsoo’s is that you have to stack it before it is doing AoE splash damage. Death’s Dance makes Dark Flight and the bonus damage from Blood Price heal you so you can pretty much leave Blood Thirst / Blood Price toggled to Blood Price on unless you are in dire straits and need the bonus healing from Blood Thirst. The 10% CDR you get from it is also nice as you really don’t get it from anywhere else unless you are jungling and don’t have a mana user to hand it off too.

At the point at which you have Enchantment: Devourer, Death's Dance, and Guinsoo's Rageblade your damage and healing are pretty crazy but you will be squishy and have to be really cognizant of whether or not your Blood Well is on or off of cool down. This is where Sterak's Gage comes in as it will keep you alive through a prodigious amount of burst, and if you are alive you can heal up pretty quickly.

I you don’t need any more damage, then you can look toward some more defensive options. Maw of Malmortius is good if you are facing some big AP burst. The armor pen it provides is a also nice addition to the stats you are already packing. If you want to be ultra troll or you are getting focused down too quickly then Guardian Angel. You get more bang for your buck with if the enemy teams damage sources are mixed, but that isn’t a hard requirement as most champions either have some magic damage in their kits or they get magic damage from items they purchase (think Enchantment: Devourer or Statikk Shiv and the energized strike from Rapid Firecannon.

Spirit Visage is also a great addition if you feel like you have enough damage but need more healing and more MR and/or you are dying because you abilities aren’t coming off of cool down quickly enough. Between it and Death's Dance you will have 20% CDR.

Titanic Hydra is also beastly for more survivability, more attack damage, and even more AoE damage, plus the active counts as an auto attack reset which comes in handy when you need one more auto attack to kill off your target.

As far as your jungle item goes, if you think you'll have trouble sticking to your targets then Stalker's Blade can be a good choice. If however you figure there will be someone that you will need to be dueling later in the game like a Fiora then I would recommend Skirmisher's Sabre.

Tailor your boot choice to the game. If they have a bunch of slows the Boots of Swiftness are the way to go. If they have two or more forms of hard CC e.g. Rune Prison + Dark Binding then you will probably want to pick up Mercury's Treads. If their team is AD and autoattack centric, the Ninja Tabi would be appropriate. If you are really riping it up, then you might even consider getting Berserker's Greaves but I feel that as a potential primary or possibly secondary form of initiation Aatrox does appreciate some defensive stats and you will already have a huge amount of attack speed from the combination of your kit and your item build.

I'm still fleshing out the final build and will probably always have some variation from game to game, but so far I feel like

Skirmisher's Sabre + Enchantment: Devourer
Death's Dance
Guinsoo's Rageblade

Are core. With these items in hand you are definitely setup for split pushing which might be the tactic you use if you weren't able to snowball your lanes. And if you choose to team fight as long as your Blood Well is off cool down you are going to be able to get in and mix it up which is what you should probably be doing if your team is sufficiently ahead and is not outright outclassed by the enemy team in terms of team fight prowess.

The other items are debatable, but if I could get away with it my only "defensive" item would be Sterak's Gage.

So a full build might look like the one depicted above.

Skirmisher's Sabre + Enchantment: Devourer
Death's Dance
Guinsoo's Rageblade
Sterak's Gage
Boots of Swiftness
Titanic Hydra