I'll cut this short. This is a build crafted for players who wish to master using the most(probably) versatile champion in summoner's rift.
~r8 it 8/8 no h8 m8. If you had fun or success with this guide, upvote. Constructive criticism will be tolerated, though if you find this guide boosting you to bronze 5, it's your problem*.

*Disclaimer: The author of this guide bears no responsibility for unfortunate game losses, rage-quit, scrubs complaining about the OPness of this build, or any other negative results including but not limited to being stuck in elo hell from the use of this guide. If you tend to blame others for your horrible elo or if you are easily offended, stop reading this guide now.
However, if you find this guide to be useful/awesome, an upvote will be appreciated.:P

Pros:
-Has an escape with a relatively short cooldown
-Has a stun which negates AA damage for a VERY long time
-Probably one of the best duelers
-Hard to counter build(deals both magic and physical damage)
-Beast with CRD
-Ult turns you into a tank
-AA reset with extremely satisfying sound when hitting turrets
-Can be built into ANYTHING and still be effective.(Burst Jax, Sustain Jax, Tank Jax, or even the ultimate omnia rotundiora Jax.


Cons:
-Weaker early game as compared to other champions
-Item dependent
-First 2 core items are expensive

1st:Ignite(More kill potential and turret dive deterrence)
2nd:Teleport(I usually take this spell for all occasions)
3rd:Flash(another Q with a shorter range and huge cooldown, but more reliable.)
4th(situational): exhaust(against other OP champs such as Riven)

Don't question, JUST DO IT.

Take E first for the stun and easier level 1.
Q next for a convenient escape or gap-closer.
W last and also to be maxed for DMG^ CD down.

Tips:

Upgrade your trinket ASAP at level 9 for vision+escape. It's also the most cost effective item in the game, no reason why you don't buy it.
Good opponents will try to bait your E, so always save your Q while laning.
Q MAY be maxed first in certain circumstances, such as when you are being camped by the jungler or losing the lane in general. It has a higher cooldown than W, but it's still a 6 second CD escape at max rank without CDR.
Mastering AA reset is key to winning trades.
Knowing when to splitpush or group is essential when playing Jax, as you serve as both a distraction(split pushing) and someone who engages(Ult+E+Q).

~The next 2 chapter are also for jungling.~


Playing as Jax jungle can be rather interesting. His kit makes him a strong jungler, with Q providing a gap-closer, W providing fast clearing, and E negating damage from jungle creeps.
However, he only has one CC when ganking, and it can be exploited by a good laner. He has synergy with champions who have some form of lockdown, but je is not the strongest among junglers, especially during the early game.

I would usually start with the side of the jungler with my bot lane(either golems or gromps) as the waveclear is faster for 3 person attacking as compared to 2.

Option 1: Start gromp and smite at 1:55 (the time gromp spawns), take W and take blue, proceed to golems and then red buff and recall.

Option 2: Start golems, take red, take blue, take river scuttler and recall.

Repeat,prioritizing the river scuttler at the bot lane, and out a pink ward in the bush behind your red buff.

Ganking

:
ALWAYS go in from the back, walk to the opponent and start smashing his face. When he flashes or if your team is at a very low health, you may E and Q to secure the kill. Rookie Jax players tend to engage with E and Q to the enemy, or Q>E. What you should to is to save your Q for when the enemy is escaping. It's okay to walk in and E, but NEVER E, walk in and Q, hoping to stun. You will just lose you only CC and gap closer.


Itemization:
Get devourer with chilling smite, tell your team politely that you would like to farm and you will only gank if there is an opening. Once you've got your devourer(first item, don't buy boots before you get devourer), farm river scuttlers and ping to take dragon, giving you the fastest stacking.
Once you got your devourer, buy ruby crystal or phage for additional health, and then buy vamp secptre to be upgraded to botrk as your first item.
Follow up with boots, tri force, forzen heart and you should find yourself scaling well into lategame.