[*] Your passive is not as good as you think and it will not protect you if you tower dive
[*] Keep your passive full because it give you higher attack speed, and most importantly is to farm and stop farm.
[*] About 15 minion kills is a kill, so focus more on farming; and if you overextend then stay in the push and passively farm with your E, or go take something from the jungle and return.
[*] If your Health is low and the enemy looks hungry, go into the jungle, and get your health back. Usually if they can't satisfy their greed on you they go for the tower. Surprise them by ganking from the river or in lane and snag a quick kill.
[*] As people start getting better and better at Aatrox and realizing him more they'll look at your passive to see if its silver (it might be another color I'm color blind). At this point gank people when your passive is about to return so that they attack you thinking its down.
[*] Even though you are an animal, your mid and adc are more necessary then you are, if their good give them the kills and if their farming give them that too, we all do our part.
[*] A siege minion is an important thing, stop someone from getting theirs and get yours, and you're ahead.

With Aatrox as popular as he is, expect many mirror lanes. Aatrox is a great pusher mostly because his life steal and harass makes the enemy unable to farm; I'd say he excels at shutting people down. Keeping his passive up early is the main key. With the dorans shield your passive health regeneration, you should be able to spam your E and regenerate all of your health by the time your E is back up for use. Spamming your E keeps your passive from all the way decaying and by the time you reach your lane you should be full early game. Play passively yet be very aggressive lots of farm should keep them from attacking you but not full on engaging should make it so they still try to farm. Allowing them to farm will prevent you from pushing and overextending and getting ganked. Make them feel comfortable in the lane so you don't get ganked or hugged, however at level 6 harass them down and go in for the kill.

After one of the towers has been destroyed that's usually when roaming starts. If you destroy their tower then roam but be wary because their top will be hunting down your tower. I suggest starting your roaming with the jungle. Grab farm and get gold while keeping your top safe, then once you have the bloodthirster and most of your frozen mallet ganking will be possible.

You can be a very essential aspect to team however know your limits. If you're the best initiator then wait for the perfect time. Mostly when all of them are grouped together. With your ultimate you get some amazing range so killing that adc is a very crucial part, but before you initiate make sure your teams ready and with you, as well as their health pools, yours might be high but your teams might not. Overall Aatrox is an all in champion and a poke war isn't his strength. CC can wreck Aatrox. He can't use his abilities nor lifesteal so be careful to avoid it.