Hello,
my name is 7uvan and this is my first guide attempt, so keep that in mind. All of the stuff I am about to write is subject to changes.
Jax is one of my favorite champions. I discovered him about half an year ago and I have been playing him a lot ever since. Although most people would say he is a solo top laner, I have been testing his jungle abilities and I have to say he can be a great jungler with great sustain, huge amount of damage throughout the whole game, incredible ganker and much more.
I really hate long guides, because if I want to play a new champ or try something new, I am probably not gonna waste half an hour reading thousands of chapters with pointless stuff, so my guide will be shorter than most others, and I am only going to say why I made a specific decision.

So, lets start with the runes. The armor is important because your lvl 1 sustain is not that great, magic resist will help you with your sustain in ganks/ teamfights, you need the hybrid penetration because jax deals tons of ap+ad damage and he scales on them both. The AD quints are gonna help you with your jungling a lot.

I have tried lots of different masteries, 30/0/1, 21/0/9 even 0/21/9 and 9/21/0, but i think the best option would be the 21/9 masteries. They give you a lot of damage, with 21 points in offensive masteries, and as a bonus you get that important sustain early game with the health and armor bonuses. If you are playing versus a team with lots of AD and you need a tankier jax, you could go 9/21 and be incredibly tanky after level 6, but this is not what this guide is about.

You should start with Cloth armor and 5 pots. If your teammates are helpful, you could start with 3 pots and a ward, which is pretty aggresive at lvl 1 and could help you get first blood. If you are just going to get a bit damage on blue, buy 5 pots.
As soon as you can, get the cloth armor upgraded to Madred's Razor. Do NOT upgrade that item further, you need the gold for more important items, like boots, which you must buy next. If the enemy team has a lot of CC, go for Mercury's Treads, if not you have to pick between Ninja Tabi and Berserker's Grieves. I usually pick the latter for that bit of attack speed, which stack really nice with the rest of the items of this build. Ninja Tabi are great for ganking the bot lane, because they lower the damage from basic attacks by 10% AND give you a bit of armor, which helps you overall to survive. After you have upgraded your boots, you should rush into a Guinsoo's Rageblade. This item is perfect for Jax, because it increases his attack damage and ability power with every ability or basic attack. The attack speed stacks with your passive (and with your berserkers greaves) AND you even get some more bonuses when you get under 50% HP, which is great for towerdiving. After that go for Sheen -> Phage -> Trinity force. You could start with Phage first (most top laners on jax start with it) but i prefer taking Sheen first. Dont start with Zeal. After you have bought Trinity Force, go for Hextech Gunblade, preferably starting with bilgewater cutlass. If it even comes so late into the game, buy Black Cleaver/ Warmog's armor

Your main ability is your W. It has really low CD, and gives you a lot of damage on your next attack. Pick W at level two and max it first.
For jungle jax, start with your E, then go for W at level two and again at level 3. If you want to make an early gank, pick Q instead of W at level 3. I would advice to kill the jungle until your blue buff is gone and then start ganking. Get your R at level 6 and max it whenever possible.

Pros:
-Incredible ganks
-Immense counter-jungling abilities after level 6
-Perfect towerdiver with his ult
-Can take down turrets really fast after getting Guinsoo's Rageblade
Cons:
-Not that good cleartime (and sustain) until level 6
-easily counterjungled early game

Ghost- gives you decent movement speed which is vial to ganking. I prefer it to flash because you can stun and run away if some1 is chasing you, or you can run after their flash and jump on the enemy with your Q. it happens often to flash and then they flash out of your reach and you have wasted your skill for nothing :)
Smite-if you dont know why I picked smite for a jungle jax, then you should consider leaving this page.

Jax needs to feed. that is your main priority.
In the jungle, start with wolves, then go for blue, then wraiths, then wolves again, then golems, and last but not least, red. after that go for wraiths and wolves again, then recall, buy some items and go for a gank. You should always go to a lane if it is unoccupied, f.e. an ally is slain or has recalled - you can protect your turret, last hit some creeps, and take good exp and gold. do not push the lane (with the exception if you could take down a turret) !
take dragon whenever possible and always have a ward there. You could solo dragon at about level 10-11, but always have the enemy jungle warded. no vision=death.

This is actually all you need to know about jungle jax. The rest comes with experience and after having played a few games. I would like to ask you to comment with your opinions and not just downvoting the build. I will add some more info, maybe a few more builds (f.e. tanky jax, or counter-jungling jax etc) if this one gets a decent rating.
So, have fun!