I know that the most known build for Aatrox is like a bruiser tanky guy but the title of the guide is Assassin of Carrys so the build is for an assassin like Zed. Also in the mid lane your trades will be very short (the ap carrys will do all their abilities and then they will try to walk away) so the Blade of the Ruined King isn't the best choice for your first item (the Blade of the Ruined King is better in long trades). But I will also try to put more items build variants.

Wrong item build. Only with a Warmog's Armor you will be too squishy because sure it gives you a lot of health but think how fast can someone with Blade of the Ruined King chunk you down(especially a Vayne). Aatrox is more of a splitpusher because of his Blood Price , Blades of Torment to destroy minions and Massacre to kill turrets. I recommend you build 2 defensive items, Randuin's Omen and Spirit Visage-> Randuin's Omen is great if in a teamfight you want to face the enemy carry and Spirit Visage is good against AP champions and you gain that heal increase which is really good on Aatrox. Other than that, I usually build Blade of the Ruined King and after these two defensive items either Zephyr or Bloodthirster. Why? Simple, Blade of the Ruined King gives you a good amount of AS and a decent one of AD, so with it you can chunk throw minion waves(you can do it with Ravenous Hydra, a little bit faster) but nothing compares to chunking down turrets with a Blade of the Ruined King and Massacre.

It's what I'm here for Brother

Ok thanks for the good criticism man, i will try to improve in all of that :D

Also, its spelled "assassin".

If you don't have any crit items, then you don't need points in Lethality and Frenzy .

List off your items. Why do you take them? Which should you take first, and Why? Which ones are more cost-efficient? Also list some situational items.

Be sure to fix those spelling mistakes in your "champions around aatrox" chapter. More explanation needed in your masteries and runes chapters. More information in gameplay, like mid game, late game, warding, etc.

Try to list matchups for Aatrox. See who counters him and who doesn't.

Also turn on Comment to vote if you haven't, so you won't get anonymous downvotes.