Your main focus when building Aatrox is to use him strictly as a top laner (he's decent in the mid lane though I've had a couple good games). So with that you can use him in two ways: ad or tank (sounds weird but yes its true). He mostly is built to attack, so your number one priority is to build high end attack items, up to four in the late game. I am not putting up any runes or masteries because it varies person to person, but I mostly try to go for a attack boost if possible. Doran's blade is your best friend in start, because it gives you the tiny health boost, plus a decent starting attack increase, and gives back 5 health per hit. You should have enough gold for one health potion and one trinket of your choice.

Assuming you farmed correctly and have not been killed, you should be able to do your proper build. Start with the long sword because it helps with attack and will help build into all your late game items. Focus solely on attack. The B.F Sword will give considerable damage, and Tiamat can help with health regeneration and its active skill is very useful for large groups of minions. Buying Beserker's Shoes will help with your attack speed and your movement speed, while Nashor's Tooth can help add to the attack speed.

The juicy part. If you're doing well then this next part should be easy. If you continue to build attack and attack speed, then you should be able to pick up Blade of the Ruin King and Bloodthirster because those two weapons gives you everything you need for your late game: damage, speed, and life steal. If you are having trouble, try to pick up Bloodthirster first: the more you kill, the more bonus damage and life steal you get, up to 30 stacks. This will help you maintain your health and deal out great damage against monsters and enemy champions. If you haven't grabbed a defensive item, do so. Randuin's Omen is a great and probably the best defensive item for Aatrox, because it gives great defense and a great health boost. Make sure to upgrade your attack speed item (I tend to do this last because your attack speed will be high regardless).

The biggest threat when using/facing Aatrox is his high attack and attack speed. When Blades or Torment gets activated, followed by Dark Flight and Exhaust from you Summoner Spell and you can create major wreckage; add in Massacre right after Dark Flight and you are sure to deal out major damage. His late game is impeccable, but it is his early game that worries many summoners. He is not the best in the beginning (mostly due to the lack of a good starting armor, which starts off at a mere 18). In my opinion, he has to be built one way or another; you can't mix both. If you plan on using him as a tank than build him as a tank, and vice versa when going for AD.
Remember that your passive, blood well, also provides bonus attack speed the higher it is. So when going for attack speed, find an item that can also help with cool down reduction.