So, i started playing Aatrox a lot, i tried a different variants of builds ( runes or items))
And this one suits me well and i will explain why, Aatrox is fun champion for me, he brings destruction and chaos into team fight if opponents don't know how to outplay him(you)
I chose electrocute because it has nice additional damage if u can land the combo, combo is up to you, he has a lot of variants what i don't need to explain i think, but if u need it just combine ur E with Q, add som AutoAtack and then destroy enemy with Q.
As first item i chose Yomuu's Ghostblade, this item provided as very useful, u can chase up your coward enemies with Yomuu activation and ur Ultimate. it will give you a huge amount of Movement speed, or you can just use this combo to fall back.
Next item u want is Black Cleaver due to it's passive that gives you movement and enemies will recieve more damage from ur hits.
Those are main items that i think u need to have in this order.
Next items choice depens on game situation, if u want to deal more damage get deaths dance, if u want to for example, hold longer in fight or facing multiple enemies, build Steraks Gage. Next useful item is spirit visage, it will give you an aditional healing so it will come in handy, you build Maw too for it's magic shield, but spirit visage suits better.. If u are facing AP team u will need both of these items. If u face Ad team u need randuin's omen with thornmail, i'm talking about full ad ap team.
Sometime you can even ignore steraks gage and move on on other items, but i like steraks gage due it's passive, it will give you the oppoturnity to comeback in team fight, to retreat, to stand 1v1 against stong enemy, jump in fight.
I almost forgot let's talk about spells a bit, if u don't have other teamfight startes u can just rollout from bush or just from front with yomuu and your ultimate, use your Q to slow them down, use your combo ( E dash using Q ) to surprise running enemy, it will cc them and mostly they just can't run away if u have your team behind you. At start of the game i suggest to use your Q to have some advantage of farming, if enemy will approach low health minion just pre fire your Q, if you will miss it he will at least miss the minion and lose gold,exp. You can surprise enemy by using your combo Q+E from bush or just when your are farming.
If someone did read this thanks you for reading my guide, if u have any question i will answer them gladly. Sorry for quite mess in this text.