This is my Aatrox build. Aatrox in mostly an auto-attack dependent champion and needs lots of life steal to make up for his health-cost skills. For an addition to that he needs attack speed and damage so we will go with this for now.
If you go TOP, try to max out your life steal AKA W first, so you can stay in your lanes for longer.
Do not use your E without a reason, you will only lose HP. When jumping/using Q try not to get hit with CC and try to aim with it. When going top, definitely get Warmog and try to get Randuin, since you might as well be your teams only tank. If not, you don't have to be so tanky, since someone else is as well.
If you actually do JUNGLE, get life steal still, but you don't have to get Randuin if you don't want to. When ganking, try to use Q to get over obstacles and catch them with a surprise, afterwards use E fast.
REMEMBER; You decide what to get AND READ ALL THE NOTES!!!
I explained the items in the build section when you hold your mouse on the notes.
You should know when to use them, and why but I will explain once again.
Berserker's Greaves - Is a good item for attack speed early game.
Ninja Tabi - Needed if you are against high AD and need to tank.
Blade of the Ruined King - One of the core items, gives life steal and high attack speed.
The Bloodthirster - Is great for its life steal and damage, which is why its one of the essentials.
The Black Cleaver - You only want to use this item, if you are against high tank champions, and need to deal damage.
Zephyr - This is a good item in my opinion, because it gives you movement speed bonus and tenacity, which makes you be able to escape and catch up to enemies fast, it also gives you attack speed which you need.
Ravenous Hydra - This item is great for farming and in team fights, it gives you life steal and attack damage.
Frozen Mallet - Gives you tons of health, slow effect and attack damage. If you need health and want to still deal damage, I recommend this item.
Maw of Malmortius - If you are against high AP and still need to deal damage this is a great item, for that it gives you a shield as well.
Warmog's Armor - Warmog's gives you the most health of any items you have, and health regeneration which is great for tank.
Guardian Angel - This items makes a good combo with Aatrox's passive Blood Well.
Randuin's Omen - Strong tank item, gives you health and armor, can slow enemies for your team mates, or yourself to finish the kill or run away.
Infinity Edge - This item gives you lots of AD and 25% critical chance, this means if you fuse it with Phantom Dancer, you deal lots of critical hits. As an addition this items makes you deal 250% damage instead of 200%. Using this can make you deal lots of damage but this is risky.
Phantom Dancer - Boosts your attack speed and gives 30% critical chance, fused with Infinity Edge, makes your champion dangerous. Using this items instead of Zephyr makes your movement speed slower than with Zephyr but only a bit.
But experiment it yourself at first against bots, for that you are the best guide of yourself, no one else is. Let me correct that, experience is.
It makes you jump to a distance near where you aim. This ability deals damage and knocks the enemy up ONLY if they are in the circle of your jump spot! It is a good CC ability to gank people with, but you can use this to run away from others and dodge skills shots if needed. This skill costs nothing, so use it as much as you want, but it has a CD so remember not to use it for nothing when you know a team fight is coming.
Blood Thirst/Blood Price
It has a passive and an active mode. The passive heals you every third attack depending on what level the skill is, and how much damage you deal. The active one deals bonus damage every third attack but also makes you lose lots of HP so use this with a risk.
Blades of Torment
Is another CC attack which deals damage, but it costs health to use.
It sends spikes towards the enemy and deals damage to anyone it hits in the range, and it also slows them down for a few seconds. Use this when you want to kill someone who is running, or to run away when they are following you.
Massacre has no cost and deals damage to the enemy depending on the level of the skill. It also deals magical damage so if you have AP, it counts as bonus. While active, this skill gives you attack speed as well and 175 range, which is a lot for a melee character. You should learn when to use this skill, and turn the tables to kill the enemy.
Whatever HP you lose, it gathers up and when you die, you go into stasis and regain that amount of HP back but this passive has a CD.