Firstly, I'd like to say hello, and welcome to my guide for critical Aatrox.
Critrox is a build that I came up with after realizing that Aatrox has pretty bad teamfight and late game potential. He can do damage, but most of his normal playing style is based off of dueling and life stealing his way to victory.
I tried to change that in this build by using only his Blood Thirst for early game sustain, and passive for attack speed. Even without items, Aatrox has a very potent early game, and can beat most in lane without them.
The only problem I've found with the build so far is his inherent weakness to hard CC, and that his early mid-game is a little weak, mostly during the interlude between getting avarice blade, zeal, and dual dorans, and picking up the infinity edge. Aside from that, this build should make you something to fear during team fights, and even more so during duels.
In addition, even without smite, Aatrox can pickup dragon and Baron very easily early on. Once you get a statikk shiv and level 8 or 9, dragon should be pretty easy to puck up. Baron is a little trickier, but with a statikk, IE, ravenous, and PD, you should be good to go, as long as you're at least level 16. Be prepared to use your ult if things go badly however.

Pros:
+Great Damage
+Great for picking up Baron and Dragon
+Near infinite sustain in lane

Cons:
+Weak to CC
+Countered by poke heavy champs in lane
+Hard to keep health up in lane if not played right

-> -> ->

Taking his Massacre, as with every champ, is a must. The attack speed and range is gives him is invaluable, and the AOE magic damage is great for team fights.

His Blood Thirst / Blood Price is essential. The extra damage is allows you, and the sustain it provides, is a cornerstone of playing Aatrox.

Blades of Torment is a great poking skill, and the slow can help you to chase down and finish off enemies.

Dark Flight, while great for knocking up enemies, doesn't get many bonuses from leveling up, and should be taken second and maxed last.

The runes are pretty straightforward. They give you a little bit of defence, along with some critical chance, damage, and sustain.

This is where Aatrox really shines. His abilities work together well to be one of the largest and most effective lane bullies in the game.

First off, you're going to want to use his Blood Thirst to sustain you. Keeping your health up is the first step to winning lane. Auto attack minions to keep up your HP regularly.

The next thing to concentrate on is your Blood Well. The passive attack speed it gives is invaluable, and something that you'll really need. Use Blood Price to get health in it. Auto attacking the buff your jungler is taking 2/5/8 times then a attacking minion in lane is a good way to start this off, assuming the buff your jungler is taking is the near one to top lane. If your health is high enough, use this and your Blades of Torment to get and keep your Blood Well full or nearly full while your in lane, just make sure to switch off with Blood Thirst to keep your health high.

Once you do both of those, you can concentrate on poking and fighting your opponent. Procing your Blood Price then either using Dark Flight or simply walking up to your opponent to get close, then Basic attacking them is a simple but effective way to slowly lower their health. This does have risks however, so make sure they don't do more damage to you then you do to them. Watch out for their abilities and remember that attacking them will make their minions target you. Minion aggro can be devastating early on, so be careful. Your Blades of Torment is another simple way to whittle their health down, but it's a little harder to land sometimes then an auto attack, and missing will still cost you HP.

If you get their health low enough, and yours is high enough, proc Blood Price, jump onto them with Dark Flight, and go to town with auto attacks. Make sure to watch your health and control your Blood Thirst / Blood Price Accordingly. If they start to run, use your Blades of Torment and ignite to slow them, and kill them. Try not to turret dive, but in some cases it can pay off. It is your own decision, but be careful.

As I said in the intro, your Mid-Game with this build is a little on the weak side. Being so independent from items in the early game makes you weak in the mid, as your abilities can't carry your far enough. Unless you're completely destroying the other person in lane, start playing it a little safe and try to cs, but don't neglect poking completely, and keep them off your minions if you can. Remember, you can call for you jungler if need be.

Taking cs is the best way to break out of this weak spot in the game. Once you get an Infinity Edge, you'll be even stronger in comparison to the other champ then your were in the early game, most of the time. The damage you do will skyrocket with the extra critical chance and damage. You can pretty much force the other person out of lane if you do it right, and getting double kills on a ganking jungler and a shut down top laner isn't too uncommon for me if it gets set up right.

Once you get your IE, gank and cs until the teams start grouping. Get as far in your build as you can until then. Having a Ravenous Hydra by then will increase your damage by magnitudes, and help your team win fights consistently. Getting to full build as fast as you can makes you not only a great pusher, but a primary damage dealer on the team.
In team fights, don't be afraid to get in the middle and tear their formation apart. Having an Aatrox Dark Flight into your team after their tank initiates will send most people for a loop, and having that same Aatrox four or five shot your ADC can throw off your teams balance to the point where team fighting isn't an enticing prospect. All your need to watch out for is CC, and then your good.

As per the usual for team fights, focus the ADC first. Being the main damage dealer for the other team, getting them out of the way is a good plan. If you wait until the other team burns their CC on your initiator, you get a straight shot and can kill of their carry pretty quickly. Remember, even if you get focused before you can kill anyone, your life steal will give you a pseudo-tankiness that will keep you alive for a good amount of time, and any time their team focuses you is time they don't focus your ADC.

As I find things that need to be changed, I will update it on the guide. If their is anything that you would like me to change or look at, message me or comment. I'll do my best to keep this guide up to date and working well.