I have played as Jax for a long time and this build seems to work beastly the majority of the time. I like to play Jax as an all-around Tanky Bruiser. Once mastered, he is a boss.
Some of my recent games won stats:
17-4-16
15-0-0
16-1-2 (I got my first QUADRA KILL XD )
20-6-15
25-5-11

I suggest using the Runes I have used on Jax.. the armor, magic resist, and attack damage go nicely with him. {edit on masteries] Early game, AP is not so important on Jax for those masteries to be useful. The extra health from defensive masteries suits Jax better.

I personally believe these are the best for Jax.
Ghost: You can run away super fast from enemies about to kill you as you also use your Leap Strike to jump away. You can also chase those champions who have that last scrap of health left.
Ignite: Often with Jax, since you don't have any range, the enemy will manage to evade by flashing away, especially if you Leap Strike is on cool down. So if they flash away, you can Ghost and Ignite them to finish them off.

OK... learn how to use your abilities
1. Your passive Relentless Assault is nice... it grants you extra attack speed the more you attack. If you see an enemy near by you will want to try to attack some minions to stack up this passive so you can go in with high attack speed already. Jax is purely an auto-attack player and attack speed scales nicely with him.

2. Leap Strike is very important. You can jump...practically anywhere there is another minion, champion, or ward. Keep that in mind, especially when you are on the run when you are low on health and chased by the enemy team! You can easily evade and escape by jumping back towards your own minions and ally champions.
In the same way, if an enemy champion is on the run from you, just keep up with them by jumping on them when they start to get out of reach.
It is fun to use Leap Strike when you are hidden by bushes or in the jungle, and surprise! You just jump on them out of nowhere!

3. Empower is pretty awesome, it grants a whole lot of damage when you are fighting another champion. Even better, you can use Empower, then use Leap Strike to jump on a champion to do combined damage from the abilities. Also, if you are having trouble farming alongside an ally, you can use Empower to grant yourself the last hit. Also can be used to get the last hit on a champion.
However, it is not always the best idea to use Empower and Leap Strike together, since you miss out at a chance for an extra hit. If you time it correctly and use Empower right after an auto attack, you basically get an extra hit.

4. Counter Strike is much more powerful than you realize. If you want to jungle a bit in the beginning, you can easily do so by attacking the monsters, using Counter Strike to dodge their initial attacks, and then continue to destroy them. Do not waste your mana by using Counter Strike though, only use when applicable.
Don't use it on minions in early game. You can use it in later game, when just one use of it can kill a whole bunch of minions, to save time.
So..how to use Counter Strike against champions? When fighting against another champion, I usually: start up my Counter Strike, then Leap Strike onto them, Empower immediately after the jump, then they will get stunned due to Counter Strike. If they are stupid enough to keep attacking you, use Master-At-Arms as this will render a percentage of their attack damage and ability power useless for 8 seconds. If they haven't died yet and realize they need to back away, you can Leap Strike and Empower again to finish them off.
You can also use Counter Strike, if you are on the run, to dodge attacks, and then you can use Counter Strike again to stun them a bit early, as you continue to run away.
You can also save an ally's life by using Counter Strike as the enemy champion tries to kill off your friend. It stuns them and gives your friend a better chance of running away. Jax Saves the Day :)

5. Your Ultimate Master-At-Arms is pretty nice, just grants bonus armor and magic resist for 8 seconds. Use it when you please when fighting a champion, preferably once they commit to attacking you. Also when in a team fight, if you get in between enemies, you will want to activate Master-At-Arms so that you won't be squished as easily.

1. So I normally start off with the nice Doran's Blade.
2. Then I will acquire the Ninja Tabi or Mercury's Treads depending on whether the opposing team is building mainly AD or AP. Choose accordingly, but I usually end up getting the Ninja Tabi. But don't get them if the enemy team is all AP.
3. Then start building Blade of the Ruined King by getting Bilgewater Cutlass.
4. Then start building Trinity Force by getting Phage.
5. Then finish off Trinity Force with Zeal and then Sheen.
6. Then finish off Blade of the Ruined King.
7. By this time you are pretty OP if you are still relatively early in the game.
8. Then you can either get the Hextech Gunblade or a Defensive Item like Randuin's Omen, depending on how the other team is building their champions. If you need more damage first : Hextech Gunblade. If you need more Health first: get Randuin's Omen or an item like the Black Cleaver to get you some health.
9. Your final item can depend on your enemy team's build. If they have a bunch of Phantom Dancers, well, why not get Nashor's Tooth? Or if they are super tanky, get Warmog's Armor. Just want more damage? Get Guinsoo's Rageblade.
It all depends on the game.
10. Your CORE items must be: some Protective Boots, Trinity Force, Blade of the Ruined King, Hextech Gunblade, and some item with HEALTH/armor/magic resistance.
11. REMEMBER to use the actives on Blade of the Ruined King and Hextech Gunblade !! Since Jax doesn't have a lot of CC, you will want to activate these items to slow them down.

1. Jump out from behind bushes using Counter Strike and Leap Strike
2. Immediately use Empower
3. Auto-attack for a bit, continue using Empower, and activate Master-At-Arms if they commit to attacking you.
4. Keep harassing them.
5. At this point they are probably almost dead, they will try to run away. Use Ignite, Ghost, and your active from Blade of the Ruined King
6. Use Leap Strike and Empower again to finish them off.
7. There are hardly any champions who can counter this assault. Jax is one of the toughest champions to 1v1 duel.

Jax does best when he is near the back of the conflict, as he often gets focused by the enemy. You shouldn't initiate team fights, but rather jump in a few seconds later with Counter Strike and Leap Strike to help out your allies by stunning them so that your allies can all get some damage at them. Once you are a part of the main fighting action, you should use Master-At-Arms so that you remain considerably tanky. Focus on one enemy at a time and bring them down, and work with you team.