Aatrox was my first main, it's been awhile since I've played him but he should be pretty much the same. First off-
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Never max Q or take Q first under any circumstances. You will never win a trade/all in with an early Q, it's health cost is too high
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You can take him top lane, that's what I did at first, but he get's countered by a lot of the meta top laners right now and is difficult to trade with (despite his lifesteal) just because of his terrible early game
2 (cont.) - By taking him top lane you are missing out on easily the best item on Aatrox, Devourer. It procs his W every other auto when sated, gives him the att spd needed to do anything, and is extremely easy to get due to your sustain through jungle. The only thing to watch for is an aggressive enemy jungler due to (once again) your weak early game. Taking him jungle is also a fairly reliable way to get out of your early game safely.
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Starting items:
Top: Depends on the matchup, a lot of the time it's good to go cutlass and then build a tank item. Take an early cloth + pots to speed this up as it will obviously set you behind a bit. Start Hydra if ahead as the lifesteal and wave-push compliments you greatly but only if it's a safe matchup
Jungle: Devourer -> BOTRK nearly every time. Don't forget to buy pink wards to safely guard your jungle, and don't underestimate getting those scuttles ;)
Another note: When going top lane you should be going a fighter build, that includes a deadman's/randuins/steraks/guardian angel/etc. Jungle is a much easier route if you are going a more damage-oriented att spd build
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Your bloodprice/thirst is probably the most complicated part of Aatrox simply because you need to know the conditions to use it. Once again, I'll split this for top vs jg
Tip: If you don't know already, your bloodthirst heals for 200% more if under 50% hp
Top: Early game, blood price simply costs too much hp for you to be able to trade with any. The extra damage is not worth the chunk of health you will take, especially if they damage you back. As early aatrox, before your first back/completed item your focus should be about farming and not dying. Once you have a bork or some defensive item, a good combo is Q -> Bloodprice -> basic attack -> e -> 2 more basic attacks for thirst/price. You can win most trades this way as you will proc the extra damage from your price and either have enough hp to hurt them more or heal it back. Keep in mind if you miss your knockup you will probably lose the trade.
Jungle: Start W and use your bloodprice to rack up your bloodwell, you'll need the attack speed. When you get level 2, take a point in E and use that to further fill your bloodwell while you heal with your w. Once you have a full bloodwell you should be able to sustain for as long as you want in the jungle at a constant 50% hp (give or take).
In general use of E for teamfights: Blast away with bloodprice unless being focused, then use bloodthirst. You are much too squish to time switching between at 50% health as you are most likely going to be bursted.
- General Pros/Cons:
Pros: Great jungle sustain, reliable solo objective secures (dragon for sure, baron if you have enough attack speed and a botrk), simple and easy kit
Cons: Squishy early game, difficult to come back from being behind, out-of-date kit, item dependent, doesn't provide anything unique to a team
Final Notes/Advice:
I just wouldn't go Aatrox honestly, as much love as I have for the Demon of War he simply doesn't do anything special. Shyvana is a great alternative (especially in the jungle), she does a lot of what aatrox does only better.
P.S.- Don't use Mecha Aatrox please, use Justicar or Sea Hunter (personal opinion)
Holy hell you made it through the whole thing, congrats! I hope this was helpful to you in some way, and sorry for the essay, Trox holds a special place in my heart :3
I'm out-
Slythion
[slayer-jinx-wink]
and 
are more than OK on him.
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