This is a theoretical guide to Aatrox. Every item in this build magnifies one of his aspects, focusing on sustain and never dying.

Standard Runes on any AD really, you can swap the blues for flat mr if you want.
3 AD Quints would increase your damage by a fair amount, but Runes aren't considered bonus AD so it won't increase your heal. This is why I like 1 life-steal Quint.

Masteries - It doesn't matter. As long as you don't take double-edged sword (although a good item, defeats the purpose of the build), IDGAF. Go ahead and take mana regen points, that won't affect the effectiveness of the build compared to other builds. If you are using this build in ranked, I would suggest taking something standard (as shown above) so that you aren't taken out by try-hard Fioras early game.

Movement speed can be taken in either runes or masteries if you find that having no boots is hard to stick to people (because you don't have a knockup or a slow)

Well, with any amount of common knowledge (ex: knowledge to read. Oh wait a second, you're reading this) you can read the item descriptions, but I realized not everyone does this, so I'm going to give you a brief explanation on how this gives Aatrox more sustain in an unorthodox way.

Guardian Angel

Essentially a second Blood Well, thus tripling the average life from 1 to 3. Yes, Guardian Angel gives you a THIRD LIFE. (technically it comes first before Blood Well so your second life...)
No further explanation required.

Maw of Malmortius

Gives you a healthy shield before you die, AD to increase your heal, and MR just for tankiness. The secondary passive that if you ask any average Bronze player won't know about is it gives AD based on how much health you're missing. A total of 95 AD when you're low health will greatly improve your healing ([0.25*95]2=47.5 increased healing when at minimal health).
This should be one of your last items, being quite pricy.

Spirit Visage

The must need item for all Healers, increasing your healing by 20%.
NOTE: This increases the healing of
Guardian Angel
Blood Well
ALL LIFESTEAL ( Ravenous Hydra Blade of the Ruined King Guinsoo's Rageblade
Blood Thirst
Active from Blade of the Ruined King
ANY HEALING TAKEN FROM OTHER CHAMPION (this build works great with a Soraka)

Blade of the Ruined King

Life-steal for increased sustain, attack speed to proc Blood Thirst more, AD to increase your Blood Thirst heal.
IN REGARDS OF ITS ACTIVE: when you get low on health, activate it! The average Bronze player (yes I said it again) will use the active only for the slow, but it works much better if someone at relatively high health enters the fight and you need to heal up.

Guinsoo's Rageblade

Some AD increase your Heal, AP to increase power on your slow and your ult (max 72, but hey, 0.6 and 1.0 ratios) but it's passives are amazing. Every ability and autoattack you use will increase your attack speed. No one cares about the AP.
When you get under half health, you get more lifesteal, some spell vamp, and attackspeed! These scale beautifully on Aatrox (minus the spellvamp, shhhhh). This is a very cost efficent item on Aatrox and is usually my second or third item.

Ravenous Hydra

AOE life-steal, 75 AD, health regen, you get the point.
IN REGARDS OF THE ACTIVE: don't bother using it for damage. You're trying to stay alive, the kills are just for bonus. Once your Guinsoo's Rageblade activates, or any time after it does, use it. This would assume you are under half health, so you can always use a heal. It does a fair bit of damage, and at that point, you'll have 39% lifesteal, plus the bonus from Spirit Visage.

If you play this way, you'll most likely get ahead early game (me being kinda sucky at solo laning, still going 8-0 early), because idiots will dive you under turret. YOU WILL FALL OFF LATE GAME. Just be the guy who autoattacks, dives carries, etc., but also the guy who doesn't die.

FAIR WARNING: the enemy fountain turret ignores both Guardian Angel and Blood Well. It will one shot you. Yea, I won't do that again.