This build is similar to most builds you see for jax, but instead of stacking more AP, this build stacks more health and AD. The chart at top says that with this build jax will have 3391 health, but actually once his passive and warmog are put into effect, jax will actually end up with something around 4200 health making him even that much more difficult to kill. The best part about this build is you are a DPS character so you can deal a lot of damage to both champions and towers and at the same time run into a team fight and initiate it as if you were a tank. Try this build out and see for yourself how the enemy team will begin to complain at how "unkillable" you are!

Jax's passive ability counter strike is based on dodge chance, so obviously dodge runes are the most beneficial runes for jax. Especially with this tank build, you want to have your stun available at all times. The next best runes, in my opinion, are the attack speed runes. Jax's other passive on his ult plus the effect of rageblade cause his attack speed to go up as he hits. This is nice except when you first start attacking, his attack speed isn't high to begin with. This is where the little extra boost from the attack speed runes comes in handy. Every little bit counts.

Pros:

Jax can deal massive amounts of damage to any kind of champ and is pretty resistant to thornmail.
The health boost in this build makes jax a tank to be feared, almost unable to die.
Jax's stun ability is amazing for initiating team fights and getting away.
Jax can be played many different ways and be just as good depending on the player using him.

Cons:

If you start off slow and not aggressive early game, it can make it hard to use this build.
Getting too greedy with jax can cause many unneeded deaths and not many kills.
Farming is slower than many dps characters with burst damage abilities, so kills are a must.

I have used jax to jungle when my team wanted a jungler, and it is a little more difficult than your typical jungling characters like warwick, nocturne, shaco, etc.
However, once you learn jax enough, you may find jungling to be quite easy due to the stun ability. This is where jungling needs a different skill sequence than laneing uses. As a jungler I would start with counter strike as my first ability and cloth armor + 5 health pots as my first items (the cloth armor goes into his boots). Stack the counter strike ability a couple times and empower when you cant get counter strike. At level 5 or 7 I would get leap strike for ganking out of the jungle. The summoner spells i use are smite (obviously) and flash.
The route in which you jungle is all by preference. If you can get a teammate to help with the blue buff first that is always beneficial for the mana usage while jungling.