Pros

● Heavy Damage Dealer
● CC spells: Knockup (which can't not be reduced by Tenacity) and Slow (with scaling duration)
● Can get over the walls
● No mana cost
● An inbuilt Guardian Angel
● Can reach 50% AS bonus from level 1
● Incredible sustain

Cons

● Most of the damage deat is Physical
● Can be easily countered by CC (especially AS slow)
● Relatively low base stats
● Speel cost is % HP based (10% HP for [Q] and 5% HP for [E])
● Ultimate effectivness is very situational.
● Second Skin is a piece of schieze.

Starting Agressive

Doran's Blade is very effective on Aatrox, especially if you're going for a very agressive counter-jungling and early ganks. Since it provides additional sustain and damage, together with your [W] you heal off very quickly, espcially when you're below 50% health, since it improves heal from [W]x3 times.

Starting Defensive
Very few situations when you'd be willing to start like that, but that for sure will keep you away from spamming pots, and you can keep jungling/ganking without recall, but lower clearing speed.

Starting Standart

Well, not much to add here. With your sustain and 5 pots you should have no problems jungling, additional damage from Machete will help you clear camps even faster.

Core Items

Blade of the Ruined King - Probably the best core items for most Auto-Attacking Melee champions. Greatly singergizes with your [W] Blood Thirst, provides bonus sustain, crowd control, attack speed, damage and in some cases even escape.

Every other item is basically optional and highly situational.

* Mercury's Treads - Best basic item against Magic Damage and your #1 enemy Crowd Control.
* BotRK - Your primary item, which provides additional crowd control, attack speed, sustain, damage and in some cases even escape. You benefit from every aspect of this item.
* Maw of Malmortius - Aatrox is the champ that balances his health pool on half in most situations. Since you will always stay at 50%-60%~ health, you will get additional 16-20 attack damage from item's passive. Anti-magic 400 HP shield will save you from champions that rely on burst damage, rather than long fights which you'd prefer. And basic magic resists also comes in handy
* Spirit Visage - Very few champions benefit from this item as much as you do. You will gain additional lifesteal, sustain from [W] [Blood Thirst], BotRK active spell and even your Passive [Blood Pool] will restore more health. Also cooldown reduction, health and magic resistance are just what Aatrox needs in such fights.
* Mercurial Scimitar - Rather optional and rarely use item, however since your champion lives only while fights and massive crowd control makes u very vulnerable this one might come in handy. Additional attack damage and magic resistence + "cleanse" which might turn the tide of the battle for Aatrox. However can be replaced with any other item you desire if there's not much CC in their team.
* The Black Cleaver - Additional health, cooldown reduction, damage, 10 + 25% armor penetration makes this item very effective for a melee fighter such as Aatrox. I don't need to explain why you'd need all of those stats.

Basically this build should provide you enough damage, health and magic resistance to fight heavy Magic Damage composition teams and win the battle. The rest depends on yours and your team skill.