Hello people My name's Damian and I've been playing jax for quite some time now and recently I've been playing him as a jungler wich resulted in a couple of nice games.
I was thinking of making a guide on the jungle Jax and now the remake has been done, I will present to you this guide on how to jungle with Jax:

Cons:
-Easy jungling thanks to:& - Great Ganking with +
-Great Regenerating through Life Steal and Spell Vamp thanks to
-Great teamfighter
-Great 1v1 Champion

Cons:
-Early Game not the best tank(but you're in the jungle so not really a problem)
-Mana dependend, really needs the blue buff.
-Needsto do nice damage and sustain.
-His weapon is a lantern: if he would have a big hammer would smash to much so this really isn't that much of a con unless u don't like lanterns

Okay, these aren't youre standard junlger/Hybrid carry(yes jax is a hybrid carry; he pushes,tanks,kills,bursts,and farm= Carry from my point of view) runes, but i still use them.
I find it to come in handy when jungling or beating an enemy, it gives some nice damage output since Jax has 56.3 base damage + 3 From+ 6% from =62.85 damage and our 26% crit damage from our runes and masteries =62.85 times 2.26 =142 damage from a crit.
These crits give us a faster jungle route and surprising damage on enemies wich will make them turn around and try to run since they will die anyway.

The Masteries are pretty standard: I don't like Squishies(Low hp champs) so I take the HP per lvl and flat 30 hp in defence including the armor mastery for the jungle route, the magic resist is just to give us the hp and thus a bonus.

For offence is just go for all the mayor DPS masteries: damage,crit chance,life steal,armor pen,att speed. This is also for the early game damage in the jungle and against enemies when ganking, also i find Jax to be better with more AD than AP so its like a 60-40 relation with AD as 60% and AP 40%. AP IS NEEDED!!: it's great for the third hit on youre ult passive, yourand your .
AD as main source of damage allows us to 1v1 better instead of nuking people, it's better for your jungle and the damage plus life steal gives us more survivability and i also prefer him this way.

My Items Are pretty obvious:
-Start with Vampiric scepter for life steal in youre jungle
-then boots to move around to gank + 2 HP pots for more jungling and recover after bad ganks
-Bilgewater cutlass for extra damage and the active slow
-Beserker's Greaves
-Hextech revolver for AP and little spellvamp
-Finish Hextech Gunblade: you now have you your 2 core items
-BF sword > The Bloodthirster: Lifesteal and damage, very nice.
-Phage : HP and Slow > Trinity force: Just an awesome item and really great on Jax for the hit after an ability cast, the slow is great and the movement speed is fun since I love moving fast.
-Needlessly Large Rod > rabadons : Best AP increasing item in the game for the extra damage.
-The last item is very controversial since i almost never get it done because they will surrender or we will be running down their nexus like a boss.
But people complained about him being to squishy, still with this build u have so much regen i jumped in 1v5 and healed everything and got a quadra (see comment reply link)
but for u complainers<3 i put in Frozen mallet/Warmogs/Sunfire

Leap Strike - Jax leaps toward an enemy and strikes them with his weapon.
Jax leaps to a target unit, dealing 70/110/150/190/230 (+ ) (+0.6) physical damage if it is an enemy.
Cost
65 Mana
Range
700

This skill is great for ganking,finishing,getting away,starting things,juking and evading skill shots

http://euw.leagueoflegends.com/sites/default/files/game_data/1.0.0.132/content/spell/24-2.png Empower - Jax's charges his weapon with energy, causing his next attack to deal additional damage.
Jax charges his weapon with energy, causing his next attack to deal an additional 40/85/130/175/220 (+0.6) magic damage.
Cost
30 Mana
Range
300

They have lowered this CD with the remake to make it more a last hit tool and when basic attacking, this is a very powerfull spell and we must max it first

http://euw.leagueoflegends.com/sites/default/files/game_data/1.0.0.132/content/spell/24-3.png Counter Strike - Jax's combat prowess allows him to dodge all incoming attacks for a short duration and then quickly counterattack stuning all surrounding enemies.
Jax skillfully dodges all basic attacks for 1.5 seconds.

At the end of the duration, Jax damages surrounding enemies for 40/70/100/130/160 (+ ) physical damage and stuns them for 1 second. This spell deals an additional 10/15/20/25/30 damage for each attack Jax dodged.
Cost
70/75/80/85/90 Mana
Range
300

The new counter strike is now a great ganking tool first we had to have the red buff to slow people when ganking. Now we don't need that we have our stun right when we need it.

http://euw.leagueoflegends.com/sites/default/files/game_data/1.0.0.132/content/spell/24-4.png Grandmaster's Might - Every consecutive third blow deals additional magic damage. Additionally, Jax can activate this ability unleash his inner strength, increasing his Ability Power and Attack Damage for a short duration.
Passive: Every third consecutive strike Jax deals 100/160/220 (+0.7) additional magic damage.

Active: Jax's gains a burst of power, granting him   Ability Power and   Attack Damage for 6 seconds.

(These bonuses are equal to 25/45/65 + 20% of your Ability Power and bonus Attack Damage respectively)
Cost
75 Mana
Range
100

His ult now gives bonus AP and AD this is great for fighting people and you should use it a much as possible.

http://euw.leagueoflegends.com/sites/default/files/game_data/1.0.0.132/content/spell/24-5.png Relentless Assault - Jax's consecutive basic attacks continuously increase his Attack Speed.

This is his new passive, i'm quite happy they changed his passive to this.
It helps more with our jungling and early ganks and we still get enough HP from our items.

I choose this sequence because Jax is best played as a Hybrid Meaning both AP and AD so we max our , Then Taking whenever possible. and maxin last
We start our game with for our jungling then we take for more damage then take and follow the order as seen at the beginning of this guide.

For Summoner Spells i use Smite and Ghost These are great on Jax since he has a build in Flash , i've consider't to using Exhaust instead of Ghost but he needs the movement speed to get to lane faster when seeing trouble escaping bad situations when theres nothing to jump to or to just get faster to where u want to go.

I will make a Vid of This but i haven't got that kinda time right now so i will add it later.

To jungle good u must remember that Blue and Red Respawn every 5 mins keep that in mind for maximized jungling.

Jungle route: Start @ blue make sure someone pulls it and if it's possible ask for damage from the top/bot lane resulting in losing less HP.

When done with blue you will be lvl 2 and i can do the wolves pretty fast( Spamm your skills as much as possible). After wolves kill the wraiths then start doin red buff after that do golems, Recall buy boots 2 HP pots and start ganking

*(u can gank before boots depending on the lanes and your HP)

repeat this order as many times as you like untill your team starts to go into teamfights and pushes mid lane.
*(Gank as much as possible especially top and mid since they're solo)
**(Ganking will be explained Below)

People always ask me why dont i getMadreds or buy , well i already got a lantern and jax doesn't need jungling items since he has enough damage. It's a waste of inventory space.

I will also make a video on Ganking but i still haven't got that time and will be added later.

Ganking: Ganking requires some expierence with jax because u have to time skills and watch a lot of things a the same time.

The main idea is to stand in a brush near the lane tell youre teammate to get ready, once he has confirmed he is ready pressa little buff will appear and a timer will run out, when close to the end press then on your enemy and the stun should pop up, !!! THIS REQUIRES PRACTICE AND WILL NOT ALWAYS WORK !!!

You should be able to gank anyone like this but be aware of wards, and when youre opponent has a stun try to get the timing in a way that it'll pop when your ends to prevent from getting stunned yourself

The farm is very important on Jax because we use the most expensive items in the game try to farm as often as possible: when u ganked a lane, when someone recalls, in your jungle and the enemies jungle.

i mostly end around 200-300 CS and still haven't got my last item most of the time, so keep
up the farm !!!

To sum things up Jax is great jungler and ganker wich relies on his skills and basic attacks, He needs a big CS and will do enough damage to kill people and carry games.
although he still needs his team, he can't 1v5 the entire game.

I've played Jax This morning (the remade version) and we won at 25 minutes and i went 3/1/3.Died because of a Karthus Ult > Jax Game

Here i was trying out some build as Jungle Jax > Few jax games

So Please try it out before Commenting and please leave any questions or comments,
feel free to correct my english in any way since it's not my mother language.
Thanks for reading
Greatings Damian