I decided to upload my Jax build here because it's brought me great success game after game. Jax is the ultimate 1v1 champion and a great carry hero, and because this build gets him pretty tanky, you can win most 1v2s as well. If you give good feedback, I'll even update this guide.

I build defensive masteries. The extra dodge is perfect for Jax because he's built heavily around dodge. Ardor's extra attack speed and ability power (which are for some reason in the "defensive" section of the mastery tree) are nice. I take Exhaust and Cripple, but some people don't use Exhaust; obviously you shouldn't necessarily follow a build to a T, and if you don't use Exhaust, don't get Cripple.

Because Jax's E and R are both based off of dodge, dodge runes are important. Currently, my own rune page is a menagerie of 1st- and 3rd-level runes, with the exception of my yellows; they're all 3rd-level dodge runes. There was noticeable improvement as I went through my rune page buying the yellows and they're key to Jax gameplay. I'm still buying my 3rd-level reds and blues for this build, but these are more flexible to your personal preference. I like these because armor pen. reds eat through tanks and what little armor non-tanks may have, and magic resist blues help defend against abilities. Because of all the dodge you end up with, you don't need to worry as much about autoattacks so armor runes aren't needed.

I take Exhaust because it makes getting kills a lot easier. Cripple means that if you're 1v1ing and losing (which really shouldn't happen), you can pop it on an enemy to lower armor and magic resist and give yourself an advantage. Early game during the laning phase, if you and an ally go for a kill and fail, Exhaust is a second chance. Even if you fail to get the kill when Exhaust is up, the slow will usually give you enough time for your Leap Strike to come off cooldown. Ignite is great for laning phase because even if you fail with an Exhaust, pop ignite and your foe goes down. If they pop a health pot, drop Ignite on them and cut their healing down. You can force enemies to B this way. Plus, if you hit a Mundo or someone with similar healing abilities, Ignite can give you an opportunity to finish them off before they can heal themself.

Most people rush Jax's Leap Strike because they like the damage. I prefer to rush Counter-Strike first. The higher dodge means that in early teamfights, as soon as your opponent has used all of their abilities and they're on cooldown, they have to autoattack you or be slaughtered. However, their autoattacks lead them to the slaughter anyway; your dodge will pop and it will become easy to spam E during the fight. It makes harassing easy as well during laning. Every time a wave of minions comes, run to the front so they attack you. Your dodge will pop; hit a minion twice if you're 6, Empower+Leap Strike on an enemy, pop Counter-Strike, and smash them with your ult's passive. Because you can harass like this, it makes zoning easy. Rushing E also improves Relentless Assault's defensive abilities which is essential for avoiding ganks and surviving teamfights. Remember to use your ult in teamfights; where dodge will help you defend against autoattacks, Relentless Assault is perfect for avoiding getting owned by abilities.

Jax isn't a very good farmer until he hits 6. He becomes excellent at that point and you should farm as much as you can from 6 until midgame to lategame's push-heavy and teamfight-heavy gameplay. Jax is easy to harass with, especially after 6 and especially if you rushed Counter-Strike. During every wave of minions, walk in front of yours so enemy minions attack you. You'll dodge one or more of their attacks 90% of the time. Immediately Empower+Leap Strike+Counter-Strike your enemy. If you're six, hit minions until the extra magic damage comes to the next hit and do your sweet combo. The smash will deal a great amount of damage. If you don't have the opportunity to get Counter-Strike up, Empower+Leap Strike does decent damage even with both at level 1. You can usually just jump your enemy and run back, as most players automatically retreat when attacked. If you kill one of the enemy champions or if they're both heavily damaged and you aren't, zone them. Zoning is where you run behind their caster minions and farm, preventing them from getting experience and gold from minion kills. Remember to conserve your mana; don't waste it all harassing or if a fight breaks out you'll become useless.
In the even that a fight breaks out while laning or a gank is attempted, remember that you can Leap Strike to friendly champs and minions. This is a valuable escape mechanism, but remember that most players don't immediately realize the need to let you jump to them. If you're in Vent, don't forget to scream "COME BACK" at your laning partner. Jumping to minions can be used to chase, but don't do it unless you have Exhaust up and jumping to the minion will put you in Exhausting range or you won't have the time and damage needed to finish your fleeing enemy.

I start with Doran's Shield, although this first item is probably the most flexible. I pick it because I don't use a lot of mana while laning and I like the extra armor and health regen, which helps lane for a longer time. If you're the type of player who uses abilities as much as you possibly can, get Doran's Ring instead, or you won't be able to contribute much to fights.
The bit of extra armor from Ninja Tabi is nice, but it's the dodge you want. As I've said, Jax is dependent on dodge for Counter-Strike and Relentless Assault.
Guinsoo's Rageblade is great because Jax is a hybrid and it gives both AP and AD, and because of its passive, which is basically Jax's ult minus the extra damage every three attacks.
Some people buy another item and then get the Hextech Gunblade, but in my opinion, the AD and AP are great, and the lifesteal is awesome. Between the attack speed from Guinsoo's and Jax's ult, a normally small or average amount of lifesteal becomes huge. The spell vamp is excellent as well, but not quite as important as the lifesteal. An important note: remember to use Gunblade's active. If an enemy is fleeing, the slow is a huge help for chasing, or if an enemy is at low health, it often gets you that last hit needed for a kill.
Relentless Assault's passive along with Rageblade's passive means that a lot of damage will be coming from straight autoattacks; this is why I get Frozen Mallet before Rylai's Crystal Scepter. The health is awesome and the damage is awesome, and the passive means that enemies will stay well within your grasp. Rylai's is basically a Frozen Mallet that works for abilities instead of autoattacks.
Sword of the Divine is great because the massive attack speed boost combined with Guinsoo's speed boost and your ult's speed boost means that your ult's passive and Sword of the Divine's passive will both be popping a lot doing massive damage.
Some say that if you get a Frozen Mallet, you don't need a Rylai's Crystal Scepter. I prefer getting both because of the health; I find that having both makes you a better tanky DPS.

The 6th item is as flexible as the first item. You may want to throw a defensive item here based off of enemy champions, although all the health you get from your passive, Frozen Mallet, and Rylai's Crystal Scepter usually make you so tanky that nobody can touch you anyway. Here are some suggestions:
Atma's Impaler: The armor and critical strike are nice for doing lots of damage with your autoattack. Assuming Mobafire is right and, with this build, you'd end up with 4126 health, Atma's will give you 82.52 attack damage, which is an awesome boost.
Lich Bane: This item is nice if you want some more AP and mana to burn. I don't think you really need any extra mana, however; I never run out of mana playing as Jax. The passive is what really matters here, though. 100% of your ability power lategame is a ton of damage added to Empower+Leap Strike.
Phantom Dancer: If you have lots of fast enemies that get away, Phantom Dancer is a great chase item. I don't think Jax really needs more attack speed, but hey, to each his own.
Black Cleaver: If you have a full Jax rune page you may not need the armor pen. this gives, but it is suited to Jax's tastes. The stacking armor pen. rips through tanks (and everyone else), the attack speed boost is nice, and it gives a decent amount of attack damage (which will even boost your health a bit).
Trinity Force: Kind of expensive to grab late-game with only 1 item slot open. If you DO get it, I recommend you get Sheen first. The little bit of everything you get off Trinity Force doesn't make a huge difference in any one area, but it does help cover a bit of ground, and it may give you a slight edge in a certain area (health vs tanks, speed vs fast heroes, etc) you need.
Youmuu's Ghostblade: The CDR+damage is nice, extra armor pen. and crit chance is a bonus. Like I said, I never run out of mana as Jax, and Youmuu's means you can spam abilities to your heart's desire.
Nashor's Tooth: This is nice for a hybrid because it gives AP and attack speed. It cuts cooldowns by 1/4, which is pretty huge.