Hi, my name is BananaPill, and this is my Jax jungle guide.
I've been maining our Icathian warrior for a 2 years and I'm pretty sure that I might guide you.

Pros

+High Damage
+Tanky
+Very Useful in Late
+Good Splitpush
+Counters Master and other autoatack champs
+You can build wathever you want and it will work

Cons

-Kinda Weak in Early
-There are champs with better CC
-He doesn't look that cool like in Cinematic


Summarizing, Jax is a high damage bruiser with good CC and gap closer.

Conqueror with Press The Attack are best two main runes for Jax.
Of course, you can go After Shock for more defensive option or even Elevtrocute, but I don't think it really makes sense.

Back to Conqueror and Press The Attack.

First one is againist everything, it works on squishies and tanks. The good thing is, when it's stacked (you need to be 3 seconds in combat to activate Conqueror)
it (kinda) buffs your W and R's passive, so then it doesn't deal magic damage, but some of this magic damage (and physical of course) into true damage.

Now about Press The Attack
It's really simple.
If you autoatack your enemy 3 times, you'll deal more damage and make your enemy vunerable for damage for next 6 seconds.
This really powerful on Jax, especially, if we know, that our W is a autoreset, so we can amazingly fast do 3 attacks and go back.

Heroism runes: Overheal,Triumph, and Presence Of Mind.

From these three I mostly prefer Triumph, because it really helps after a "somehow" won 1v1 duel with enemy jungle.

Overheal seems very useful, but I don't really know what to think about this rune.
I will try it and then I will update this guide.

Presence of Mind is like a Triumph, but less useful (for me of course).
Why? Jax haven't long Ult cooldown so 1 part is "meh".
Second part, (giving back mana) is useful, but I prefer Triumph.

Legend runes: Alacrity, Tenacity and Bloodline.

Which one you choose depends on your playstyle, I mostly prefer Bloodline, it gives you really nice lifesteal.

Combat runes: Coup De Grace, Cut Down and Last Stand.

I prefer Coup De Grace, it helps when someone is runnig on 40% of hp and you need to jump in and fastly kill him.

Other runes are good too, which one you pick depends (once again on your playstyle).

Second rune tree would be Determination or Domination.

From Determination I pick Boneplating and Chrysalis.

When I go Determination I always go Sudden Impact and Ravenous Hunter for even more lifesteal.

As you saw above, I play maxing W and for second one I go 3 points in E and then I max Q, and for end E.
I'm maxing like that, because you'll need this stun in early (kinda) more than a gap closer.

Jungling with Jax is easy.
You start with E and when you start to do Red/Blue for a first camp you just use E and autoatack the monser.

Your job as Jax is to go into enemy team, stun everyone, tank as much as you can, kill carries and count on your team that they will go with you.

You can activate Sheen's passive 2 times with your E, one time when you start your E, and when you manually stop it.

You can easily get some stacks of passive by using W and Titanic Hydra.

You can jump to wards and use it as "reserve flash".

Use your ult in teamfights to tank as much as you can.