Jax is a great 1v1 character, he attacks fast, he attacks hard, and he can get away if the fight gets too hard to handle (which is rarely).

This is a build ive thrown together using aspects from other peoples builds and Jax' own attributes.

The build only shows 16.5% Dodge, but in game i get to 28%
My AD gets to 305
My HP gets to 3000

With this build I destroy the enemy team, I have Solo'd Baren, and I have gone undefeated.

Let me know what you think.

Marks - After purchasing Zeal (on your way to Phantom Dancer) and getting Ulti, Jax attack speed will be very high, but untill then it is important to have fast attack speed. This is why I have all 9 runes dedicated to attack speed. Ulti only speeds up your attack when actually attacking, so if the enemy team attempts to gank you, you wont have fast attack speed. This is why you need Greater Mark of Alacricy. My attack speed caps out at 2.5 while attacking.

Glyphs - Jax can be built as a tank, and with 21 masteries in Defense he can take a lot of damage. However, it is better to build him DPS (Damage per second) with high AD (Attack Damage).
For this we have items, they will be a great aspect. But in order to purchase these items you need $, and in order to get $ you need to survive.
All 9 Greater Glyphs in Sheilding make you survive against mages, such as Brand, Morgona, ect. They tend to be squishy, and if you can survive charging them, you will easily kill them.

Seals - Armor blocks damage, blocking damage is very necessary, it makes you harder to kill, since you will be harder to kill, you can 1v1, 1v2, and sometimes 1v3, with little to no difficulty. It allows you to be a greater help in team fights as well, since you can slightly tank some of the damage.

Quints - Jax's E ability (Counter Strike) is a passive ability, however, it can be activated to stun enemies and deal out damage ONLY within 3 seconds of dodging an attack. For this reason we have Masteries in Dodge chance, Ninja Tabi for dodge chance, and quints in dodge chance. This build only says 16.5 dodge chance, but in game I normally get to 24-28% dodge chance.

Jax' Passive Ability grants him 3 health per additional Attack Damage, and 2 health per additional Ability Power.

So we start the build with Doran's Blade, it gives you a base boost of 100 HP, and 10 AD which gives you 30 HP more. The 3% life steal comes in handy as well.

Next we start Ninja Tabi, Buy the basic Boots first, then the Cloth Armor.

After this go for Rage Blade, it gives you +35 AD (giving you 105 more health) and +45 AB (giving you 90 more health). It also stacks boosted ability power and attack speed (up to 8 times) maxing at +48 Ability Power and therefor 96 more health (while in combat).

Next is Sword of The Divine, by this time you should have Ulti, so every 3 attack does extra damage, and with sword of the divine every 4th attack does extra damage, boom.

Atma's Impailer - Gives you more armor and critical strike, good to have. It passively gives you 2% of your HP as AD, which then turns back to additional HP (due to passive). How much it will boost you depends on where you are with your other items/stacks/runes.

If you are strapped for gold at any time after purchasing the first few items or recipies sell Doran's Blade.

Bloodthirser stacks upon kills, giving you 60 AD (180 HP) and 15% Life Steal.

Lastly buy Thornmail, Giving you armor, and whenever you get hit it deals damage back to your attacker.

You can lane with Jax, you can jungle (after a few lvls), and you can gank.

Ganking is the most fun and effective, I recently played a match where I never pushed, all I did was run up and down the lanes and kill them, we won, I had 55 kills, no deaths 5 assists.

Hit W (empower), Hit Q (leap strike), and that already takes out much of there health, a few more hits and they are dead. If they start to get away you Q should be cooled down, and then you can catch them. Otherwise you can Ignite (for last hit) or Exhaust to get them. If minions are around they will hit you, if you dodge them your E becomes active, and then you can stun enemy champs.

Rarely activate your Ulti, the passive is much more effective, Activating Ulti is a loast resort for when they have too much magic damage and are focusing you. RUN LIKE HELL.

Also, I have solo'd Baren with this build.