I crunched a few builds for Aatrox. It was less work, as I've avoided most builds without lifesteal. It just seems like anyone would prefer building lifesteal on Aatrox, and I agree with that notion. Especially after looking at the numbers.
9/21/0 masteries are assumed. The sheet assumes 50% uptime on ult.
ASPD and AD are exactly even for BoRK, but ASPD pulls ahead with more hp
ASPD is better for BT
ASPD is better lategame
ASPD is better with dmgW at level 1
ASPD is better without ult
AD is better with healW at level 1
AD is better for burst
AD is better with ult on
According to the sheet, ARP is very good, especially later on. It means smaller heals and less damage taken early, but more healing later when you have a big LS item and less damage taken. But the E of Aatrox deals physical damage in my model, so it might be skewed too heavily.
ARP is even better lategame
ARP is better with BoRK
ARP is better against low armored targets
ARP is worse against heavily armored targets
For junglers I suggest taking the usual 3 ARP marks regardless of whether you go with AD or ASPD. I recommend against going full ARP for junglers, as so much of it is wasted against some of the jungle camps.
Personally I decided on ASPD runes for jungle Aatrox, but none of these 3 offensive runes are bad - they're all very close. It depends on where you want your strengths.
If you only have one damage item, at level 11, against an opponent with the usual average hp, average armor + flat seals + 5 from mastery and a Sunfire Cape, using dmgW, Last whisper is actually the most cost efficient choice. Even without having any other items. That was a bit of a surprise. IE comes in second, followed by BT and then BoRK. BT and IE are nearly equivalent at 9% and 11% cost effeciency lost compared to LW, BoRK losing 21%.
If you don't have a LW, now is the time. BoRK is best paired with it, second comes IE with 2% loss, lastly BT with 13% loss.
Of course, at this point, the good old IE+LW+PD is the best, followed by IE+LW+BoRK at 10% loss, then BT+LW+IE/PD both at 15% loss, followed by BT+BoRK+LW at 21% loss.
Now that is amazing, the double LS build is actually very competitive damage wise. You gain a ton of extra resilience from having as much LS as that build offers. Let me show you the numbers.
It reads "HPS with dmgW" / "HPS with healW" / "HPS with healW under 50%"
Look at that amazing healing. You're literally unkillable 1v1 against low damage targets. Hell, you could probably take two. So I very strongly endorse builds with lifesteal. In fact, I very strongly advocate builds with BoRK specifically, as it shines much brighter than BT lategame since you're not getting much crit. BoRK is after all a premium bruiser item.
If you really want your early midgame damage to shine, get BT->LW->BoRK. If you value BoRK active and/or midgame and lategame, get BoRK->LW->IE/BT. If you're only getting two offensive items, get BoRK->LW almost no matter what.
As for Trinity Force, I'm afraid it's not good on Aatrox. It's at 64% loss for a first item, 39% as second item (with BoRK), and 28% as third item with BoRK and LW.
Hybrid Defensive Items
When it comes to the hybrid defensive items, both Atmas and Maw are decent. At Slot 3, Atmas loses you 27%(at level 18 with +400 hp), Maw loses you 34% (at 75 assumed average AD from Maw)
I haven't crunched for slot 4 with Aatrox, because with that LS, getting resistances and making sure you can't be bursted is simply too good overall. I heavily recommend against going glass cannon. I can however almost guarantee you that BT/BoRK+LW+IE+PD is the best build.