I'm going to keep it short because this build is extremely easy to follow and adapt to personal play styles. Aatrox's large amount of passive self boosts and his large sustain make him the best jungler for doing solo jungle runs even at level 1.

Having built this guide over a period of 25 games i found it being effective with a total K-D-A of:
118-55-92

Aatrox jungling solo early game allows your top and mid-laners to start poking earlier in the game and allow them to soak up experience and set up ganks for you to come in and secure the kill.

(Ignore the poor job on Rengar, please)In the three highlighted games, I used the build for Pure Solo Jungle. The results are a bit underwhelming until we get to the top of the list, A custom game where i went 25-0-1.

In this Game i was able to solo Baron Nashor at level 18 (only because I wasn't sure how well I would do) however, by level 15 I had the life steal and attack damage needed to solo Baron.

By doing this, I allowed my team to continue to push mid and engage in team fights while still getting that very valuable Baron buff.

Early Game

The idea behind this build is to build to be a jungler that can solo jungle at level 1. It is recommended to start at blue and the move to wolves, wraiths, golems, and then red. this will put at level 4 or 5 and a decent amount of cash stored up.

Ganking top lane with Aatrox requires that you lead with either Dark Flight or Blades of Torment to slow them or close distance, keep Blood Thirst active until they both, BOTH, are below 50% health, ignite one and go after the other while activating Blood Price to deal the final blows. If your passive activates, return to the jungle until you are needed again. Your passive is key to winning a ganking skirmish.

Mid Game

Prioritize jungling behind enemies to box them in and slow them with Blades of Torment. In team fights, Dark Flight becomes a fantastic mass damage ability that will allow quick kills if used in succession.

In this phase of the game you should be between 8-12, go after dragon and enemy jungle when possible for gold to finish your core items.

End Game

Its probably gritty out there, your teams kills are high and so is the enemies, you want to Baron but the team keeps getting locked up in mid lane. Go ahead and Baron solo. Here is how:

-Wards should be placed at top, mid, enemy jungle entrance and your jungle entrance to keep from being ganked or having baron stolen. Have a Teemo with you for best results on warding.

-Lead off by jungling Blue, Red, and wolves with Blood Thirst on to keep your health up and still building your blood well. When you get to Baron, use Dark Flight with an immediate Blades of Torment to get a good amount of damage in before he attacks you. Activating Massacre At every cooldown for the speed boost allows for superior sustain and high damage output. Use Dark Flight and Blades of Torment as you see fit, i was kind of conservative with my health.

-Be sure to have your core items AT LEAST before engaging and use Ravenous Hydras active every single time it cools down to sustain more. When Baron drops below 50% health, switch yo using Blood Price to deal massive Damage. Start dropping hard on him with Dark Flight and Blades of Torment at each cooldown, as well as Massacre at each cooldown still, to finish him off. Use Ravenous Hydra to use life steal to sustain.

And there you go! That is how you solo baron.

After you baron, try going backdoor to go all in for the win. After Baron at level 18 i was able to go 3v1 without activating the Blood Well passive restoration. Fairly OP. Attack speed peaks between 2.75 and 3 (full Blood Well and Massacre activated) depending on the runes you use.

The runes for this build are similar to building a jungle Rengar, But there are two deviations that work equally well that i shall post.

Rune Selection 1:
Greater Mark of Attack Damage x9
Greater Seal of Armor x9
Greater Glyph of Scaling Magic Resist x9
Greater Quintessence of Attack Speed x3

-- This selection boosts early game damage and defense from monsters and top lane champions but starts to lose effectiveness mid game until the magic resist gets a chance to max out around level 16 to level 18 --

Rune selection 2:
Greater Mark of Attack Speed x9
Greater Seal of Armor x9
Greater Glyph of Scaling Magic Resist x9
Greater Quintessence of Attack Speed x3

--This simple change will effectively double jungling speed and great increase your sustain and life steal totals early and mid game and then late game will make you a very high priority target due to the large ammounts of damage you will do in a short period of time, so be careful.--

After building off the recommended Rengar-like style i noticed that Aatrox's abilities, especially Blood Thirst/Blood Price, weren't restoring enough life/doing enough bonus damage mid to late game to keep Aatrox effective and sustainable.

The masteries I recommend are a 21-9-0 focusing attack speed, ability power, and damage, then health and extra defense.

This layout may seem a bit strange and I know that there are some Ranked players saying "Why are you giving AP to a melee Champ noob?!" but when you look at his abilities, Aatrox relies on AP for a good portion of his bonus damage on most of his abilities. Trust me it works, 25-0-1 with an almost Penta-kill (Broken by Malphite) and solo Baron are my proof.

Aatrox is a low defense and health champion that relies on health for his abilities, first instinct, Warmogs. That instinct is WRONG! Next thought is build attack and life steal to have permanent sustain.... BINGO! The most effect course of action is to build attack and life steal to aid in the casting process when you get into team fights and start rapid casting abilities and use Blood Price to deal extra damage rather than Blood Thirst for sustain.

Starting items for Aatrox Pure Solo Jungle are shockingly different to any other jungler as he only uses Doran's Blade to increase his starting damage. This is extremely useful for sustain against opening blue or red. And, yes, no Heath potions. Aatrox's built in sustain is boosted enough by only Doran's Blade to be sufficient.

There are two possible End Game builds. The first is recommended to keep up with the need for more and more sustain, but if the enemy team is starting to trample yours and you need to get that damage out fast, then the second build is for you.

Prioritize Blood Price/Blood Thirst in order to build heavy sustain early for better jungling and ganking. Follow up by building up Blades of Torment in order to slow down enemies so they cant escape and you can get the kill. Finish Dark Flight last as it is mainly used as an initiating tool for most encounters and has quite a steep health cost so dont use it too much.

The recommended spells for this build are ignite and smite. Aatrox's Dark flight can be used in place of flash as an escape mechanism. This selection of power spells allows for last hit jungling and Kill Secures while ganking or laning, maximizing potential profit.

Pros

Heavy damage and high attack speed allow for higher sustain with life steal
Soloing the jungle allows for level boosting for top laners and earlier poking
Being able to solo Baron early allows the team to get that all important buff in the middle of a team fight, possibly changing the course of the skirmish
High sustain and Ability Power keep you alive in those tight situations

Cons

By not building health, Blood Wells activating effect will not restore much heath so run if you get caught off guard
The expensive build takes a lot of time to complete, jungling alone will only do so much to help out with the costs
Aatrox's high attack speed and area of effect damaging items will sometimes cause a Kill steal, so warn your teammates of this if that is something they often report people for
Soloing Jungle is always dangerous, with Aatrox's low armor and magic resist intercepting a ganking opponent by yourself will be troublesome if you aren't prepared