Jax is famous for his awesome fighter/ assassin build, and really great at solo top, and catching up. What if we used those catching up skills, and damage, and made jungle jax? I'll teach you how to be one.

For Runes, I would take Physical Dmg marks, Scaling AD. Seals, Armour Glyphs, and Physical Dmg Quintessences. Early game, you need the armour to help increase your survivability when leaping, and jungling. I play AD jax for quick Dmg, and assassination, with the and physical Dmg.
I would go for offense and defense, with offense having monster Dmg, and standard AD masteries, and armour, health, and health regen defensive masteries.
For Summoner spells, take smite and flash, increasing jungling speed, and flash while chasing/ get aways. I play Jax really defensively early game.

At Lv. 1, I'd take a point at empower: Increased 40 Dmg would help, and has a relatively low cooldown. Use it after your attack, as that resets the timer, dealing tons of Dmg.
At Lv. 2, I'd take a point at counterstrike, as it will increase jungle speed, and survivability.
At Lv. 3, Take Leap Strike, Leap at them with enpower, counterstrike, and use basic attacks for ganking.
At Lv. 6, take your ult, greatly increase Dmg with your Passive + The Ult's passive to deal 400 Dmg per 5 seconds.
At Lv. 9, you should have maxed out Empower to increase leap strike Dmg, and jungle Speed.
By Lv. 11, you should be dealing around 900 Dmg per 5 seconds, because of your drastic increase of your ult's Dmg, while greatly increasing survivability when activated.
By Lv. 13, your counterstrike only has 12 second cooldown, so you can keep stunning when engaging teamfights. Note that this will not work against heavy casters, like Syndra.
At Lv. 16, You should be dealing tons of Dmg around 1200. You can also greatly increase your team's potential with guardian angel.

First, I'd start with Cloth Armour, and 5 health pots. The Cloth armour compensates for his lack of early game survivability, as you don't have counterstrike. By first recall, get a Madred's razors to increase your armour to around 50, and deal tons of Dmg, allowing you to gank more often. At Lv. 3, I'd go for my first gank. I jump the one closest to my turret, and use counterstrike, and if your teamates co-operate, you should be able to pick up the kill. After, I'd get Wriggle's and Boots of Speed. With Wriggles, ward the river so no one tries to use a gank, as we should be heading to mid game. Upgrade to Mercury Treads/Ninja Tabi, Increasing Tenacity help with chases and getaways. You may also want to ward brushes and various parts of the map, as you can leap at them. After, I'd get Guardian Angel and Bilgewater Cutlass, now you can engage teamfights, and gain around 80 armour and 50 mage resist. Make sure you die strategically so you can revive properly. After, I'd get the Hextech Gunblade, Increasing my Lifesteal, Spell Vamp, Attack Dmg, and allows me to shoot people. After, I'd get Zephyr or Maw of Malmortius depending on the need for magic resist, or Dmg. After, I would finish with Frozen Mallet to increase my survivability while increasing my chasing potential. Use this to mop up fights, and watch their health bars drop like hell.

To succeed at early game Jax, first get the Wolves then Blue Golems, then wraiths, then golems/gank top. Go back to wolves, ward river, and take on dragon. In teamfights, be a initiator/dmg dealer. Make sure you ult when you jump in and stun the whole team. And when you fight dragon, make sure you ward the brush in the river to increase map awareness. He can kill the dragon fast with wriggle's, His ult's passive, and passive. As Jax, focus the ADC, as your the duelist, and you can win if you know how to use counterstrike. This is useful against twitch/vayne/ adcs that rely on their basic attacks a lot more than most
Jax is also good at backdooring, because of his immense speed.