Played Jax for a long time after I started playing LOL, was my first real character to play and continued playing him almost exclusively from about lvl 6 - lvl 30. Played correctly Jax can beat any other champ in the game, yeah that can be said for alot but seriously I have yet to go 1v1 against any other champ while playing Jax and lose all things being equal. Unfortunately, at least for Jax, he is not the best of team fight champs and this game is more about team fights than 1v1.

Not to brag but while playing Jax I have fought many other Jax's. Only one has ever stood up to me and that was before I was lvl 30 and even then it was a very close call, my team feeding him is partly why he did so well against me. Even then I killed him a couple of times too. Other than that one person I have yet to face a Jax that could beat me on Jax. While this is not a super accomplishment, I do know that other Jax players have tried some of the wierd builds or even make it up with false assumptions, such as attack speed, not getting rageblade, etc etc. Bottom line, this build beat them hands down no contest. So give it a try.

This is my first build so bear with me if its ugly I am not sure how to do all the pretty things like showing the spells and ability pictures etc.

9 marks desolation
9 seals evasion
9 glyphs of force
3 glyphs of evasion

pretty simple, though I admit that marks of insight could be just as well, honestly ive done it just fine, but havent tested it to such an extent as to be able to tell you which is strictly best, having moved from more ad to more ap based as jax however its probably marks of insight that are better. Either way can work though since Jax does both physical damage and magic damage.

I go fairly typical 21/9/0

Jax is a bruiser, he gets more out of straight up offense than anything else

Defense is mainly for the dodge bonus, dodgeing is important for jax not only to avoid damage but also to activate his stun, not to mention the movement speed proc from nimbleness is a good way to chase or run.

Core Items:
Doran's Shield - this gives you some much needed survivability at the beginning, Jax is pretty weak at the beginning of the game, this helps make you not so squishy at the beginning, should be sold off later in the game to make room for better items

Ninja Tabi - The only boots that should ever be on a Jax, has everything important to jax, and being the only dodge item in the game its pretty much mandatory in order to get more consistent procs on your stun.

Guinsoo's Rageblade - Gives you ability power, attack speed, and complements your ult so well its a mandatory item

Mid - Late Game Items:
Hextech Gunblade - has everything you need, attack damage, ability power, spell vamp, life steal, the active is good too, basically this is the item that starts making you really dangerous since once you power up with the spell vamp and life steal you will be gaining back life fast while dishing out the pain, seriously this item is what will let you go up against champs with half health and not only win, but win and have full health

Rylai's Crystal Scepter - Very nice item for jax, survivability, increases burst for your abilities, and slows the enemy, this is important because with your speed and them slowed it makes it hard for them to interrupt your ult powering up

Other: (At this point is when I usually sell my doran's shield and get items based on the opposing team)

So basically at this point look at whats happening, whos the heavy hitters on the enemy team, what does the team consist of, what could you use?

Depending on the opposing team

Offense:
Hextech Gunblade - hey they stack and still one of the best items for Jax all around
Mejai's Soulstealer - If your doing well and can stack it up you will get the most for your money in ability power
Rabadon's Deathcap - Especially if you power up a soulstealer this will help your ap even more

Defense:
Banshee's Veil - Good defense item specially if they have a bunch of ap based champs
Randuin's Omen - Good against other melee champs, slows their attacks down and mitigates damage
Guardian Angel - If all else fails and its a good mix of ap and ad, this is always a safe bet

Leap Strike - Jax leaps Toward a target. If it's an enemy, he smacks it in the face with his weapon, dealing bonus damage.

Note: Leap Strike is your closing ability, you do this to jump onto your target and smack them, specially useful to close on casters and other ranged.

Note 2: Leap Strike can also be used on an allied champion or minion, can be used to get away from people. I have been super low health and ran from a warwick because I kept jumping to my minions before he could close on me.

Empower - Jax's charges his weapon with energy, causing his next attack to deal massive additional magic damage.

Note: This is your bread and butter ability, this is where your damage comes from and is a major reason why I build more ap based. Basically empower, leap strike to target, then spam empower.

Counter Strike - Jax's prowess allows him to dodge enemy attacks frequently. After avoiding an attack Jax can quickly stun surrounding enemies.

Note: This is your stun, its how you stop people hurting you while keeping them steal to beat up. This is also why dodge is so important. Basically, spam this ability after you close on an enemy.

Note 2: The stun is an area of affect, meaning you can stun everything in a small area around Jax, this mainly just takes a bit of getting used to to find the range.

Relentless Assault - Jax's greatest talent lies in his ability to grow stronger with every attack. Each successive attack increases his Attack Speed temporarily, and every third blow will devastate his target. Jac can activate this ability to gain temporary Magic Resist based on his Dodge chance.

Note: This is two fold, first you dont have to activate your ult, its always active, at least for the offensive side of things. This ability is another reason you become so dangerous if you power up, you will hit fast and hard. You only ever activate the ult when your facing heavy caster damage, it can help you survive so try not to forget it.

Good
Ghost - This spell is great for chasing and for running away, just an all around great spell. Considering the fact that you should be able to beat up most champs in a fight this can help make sure you get the kill if they make a run for it, or once again get away alive if they try to gank you.

Cleanse - Once you've gotten your ult and hextech you should be able to regen health fast off of minions, and sometimes even enemy champs. However a stun can waste your ult stacks reducing your damage and healing. Since Jax is so based on powering up his ult reducing it from full power to zero can drastically hurt your damage, not to mention survivability from getting beat on.

Decent
Ignite - Good for extra damage as well as providing reduced healing for the enemy.

Exhaust - Not a bad spell, but you should be beating them up, not getting beat up

Teleport - Too situational, wouldn't recommend

Clairvoyance - Useful, but does nothing to help you that a ward couldn't do

Flash - Very useful, but I prefer ghost for Jax

BAD DO NOT GET THESE
Heal - You should be able to heal with your lifesteal and spellvamp, if not your doing something wrong and should practice more. Not to mention the fact that Heal becomes fairly worthless late game

Revive - If your dying your doing something wrong

Smite - It is possible for Jax to jungle, but basically, you shouldnt

Fortify - If your getting overrun your doing something wrong, Jax can clear minions so fast this spell should not be needed

Clarity - Jax can suffer some mana problems early game, but not bad enough to waste a spell slot on this spell, and once you get to mid game you shouldn't have mana problems anymore

Rally - No, just No

What makes Jax unique is that he powers up. His ult makes him more powerful each time he hits something, he will attack faster, this means more hits and every third hit is extra damage. Use this to your benefit, being surrounded by enemy minions and facing a enemy champion is good for Jax, specially if you get to hit the minions a couple of times before the fight with the enemy champ. Remember you will be dodging alot of the minions attacks, this means you have your stun up, the health you lose from minions will be gained back easily from the life steal and spell vamp from your hextech gunblade.

I beat many a person by them trying to attack me while surrounded by their minions, usually they regret it. For Jax being surrounded by your own minions is worse.

I wanted to address a few things that in my opinion people get wrong about jax.

Leap Strike - This is your closing ability, while it does decent damage early game when its all you have, it is not by any means your main source of damage. Honestly its main use should be to jump into the fight, perferrably at that caster/ragned thats tearing your team up.

Attack Speed - Quite simply, not needed. Perhaps it can make you a bit stronger at the beginning but with 0 attack speed runes, only 2 in alacrity in masteries (mainly coz I need them to move down the tree and theres nothing better, rageblade and your ult will give you all the attack speed you need. At lvl 18 you will have Approx 2.1 or so attack speed, plenty.

Life Drain/Attack Speed - Life drain is not dependant on attack speed, sure the faster you hit the more you will drain, however the harder you hit the more you drain as well. With this build you may be weaker early on, but your a powerhouse late game hitting like a truck and quick.

If any other problems come to mind I will try to address them here.