Hello, this is my first posted build, though I have made many that I have not posted. The first build on the right is my favorite which is a good build in every situation I have seen, the second is a bit more burst damage for those teams that cc a lot and keep you from getting stacks from melee attacks.

Jax is fairly strong all stages in the game, but the early game is probably his weakest. Be sure you keep that in mind when modifying this guide to your game play style.

Jax is one of the best 1v1 heroes, and if fed/played right you can easily win a 1v3 in there tower and cap the point afterwords (garrison on tower of course).

Future updates will include more information on how to play, more tips, hero match ups, and the much needed item images throughout the guide. I may include a update/version chapter once I have a significant update like the patch that has been released 10/18/2011.

You will notice that in Dominion Movement Speed is a very important aspect that should never be overlooked. This build acknowledges this and the two ways one can

Version 1.46.11
Ok, So.. I updated this since Riot completely nerfed the Hextech Gunblade. It is still good simply because it provides lifesteal AND spellvamp (and active slows). Because you can no longer burst people down before they can get away nor are you so survivable, so slow has been moved up in importance.
Changed skill sequence because of the lack of burst Jax has (still a lot though).
Removed 2nd build.
Changed all Quints to dodge.

Marks: I have selectedGreater Mark of Strength because they help passive (3 more HP per AD). It doesn't help as much as damage pen marks but I have noticed the best counter to Jax is lots of CC. This means not a lot of auto attacks. So having more HP is more important. If they are not constantly CCing you, you still get the auto attack advantage in damage.
Seals: Greater Seal of Evasion for the dodge. It's Jax, not much needs to be said. If you do the first build, you get about 29% dodge at start.
Glyph: I have selected 2 Greater Glyph of Alacrity for the attack speed because Jax doesn't attack fast enough for me when he gets to level 6 and gets his ult , these help trigger the attack speed boost also helps max your Guinsoo's Rageblade exponentially.
Next I get one Greater Glyph of Focus. At the beginning, Jax needs to be able to keep up with his foes and in general needs to jump more often. This has always allowed my CD at all levels to come off JUST in time allowing me to secure several kills and save my life in enemy towers.
Finally, Jax gets owned by AP carries. He can dodge AD carries and in turn stun them, but AP kinda own Jax. Greater Glyph of Shielding has scaling MR which is critical because AP are not that scary to Jax til they are about lvl 10+. This makes Jax a real toughie for all foes AP and AD alike.
Quints: I have 3 Greater Quint of Evasion for the dodge because... JAX = Dodge. Just do it.
I have one

Standard Jax mastery page, max damage where you can. Jax is hybrid carry meaning he utilizes AP and AD to maximize his damage output.
Also, be sure to get Nimblenessin the defense tree. You are a very dodge based hero, so getting the the extra movement speed when you dodge with a 30% dodge chance is great.

The goal is to get some quick massive burst damage at the start so that foes fear you from the start. But near the end game you can't just burst to kill people/ people build items to counter you. You need to start getting more steady stream of damage (auto attacking). Jax needs to win 1v1 with disadvantages (less HP/in there tower) all throughout the game. This is made possible with life steal and spell vamp as well as great burst damage, short CD and survivability.

I start with Ninja Tabiboots for dodge (yays!) and the all important movement speed! and Ruy Crystal for extra tankyness at start and a health pot.
I recommend getting one health potion just because Jax is not that durable at this level and he is a melee champ running into a 4v4 team fight.
The ruby builds into your next item Phage. You need to slow those pesky runners and keep them in a poor position.
After this you want to get more consistent damage ie auto attack damage so you build Guinsoo's Rageblade .
Vampiric Scepter for life steal early on. Also builds into next item!
Next I build Blildgwater Cutlass for that Movement Speed advantage (active).
Next you want to get the full Hextech Gunblade because you get nice life steal and spell vamp (and you can actually deal some nice AP damage with empower at this point). Not to mention the additional range on your slow (active).
AP results in your leap strike /empower combo being extremely scary burst damage.
Afterwards, you now want to complete Trinity Force so that you get more attack speed and additional burst from sheen's effect.
Now you want to get a bit more life steal just because end game is about continuously dishing out damage. This is why I get Stark's Fervor. The passive is nice too plus the attack speed.
I think that the best item to finish with is Kitae's Bloodrazor because it gives you more attack speed and nice effect for those large hp foes. But mostly, you need more damage.

I have selected the first 4 skills to support burst damage (at least one in each).
You should get drill/refinery first so that you are level 4 before you engage at windmill. This means you are a force to be reckoned with. You want to max leap strikefirst because you need it to be able to keep them in melee range and each rank reduces cool down as well as your burst damage.
Always put a point in your ult , it's your ult and is better than all the other skills.
Then max empower , to maximize your burst damage.
Then counter strike .

I get garrisonso that it is possible to tower dive all throughout the game and it allows you to defend points from a long range as well. This just causes Jax to be a force to be feared. No where is safe if Jax is determined.
Ghost really helps with the theme of this build, you need to be fast in Dominion.
Cleanse removes the one weakness Jax has. A smart Jax never dies unexpectedly if he can control the amount of CC on him. (exception, defending your points is often worth dieing for if you can hold them off for any substantial amount of time)
After the patch I think i prefer cleanse.

If you want to initiate a fight, feel free to engage creeps that aren't targeting your creeps to get a dodge. This allows a stun right off the bat when you leap strike/empower the foe.

Be sure to use garrison if you need a point and they are defending it.

once killing the final hero defending a point INSTANTLY right click on the point to start caping it. This will stop it from attacking you.

In general, don't push minion waves.

Always watch map. If you notice you are certainly going to lose a point, that means there is a weakness someone in their defenses. Cap something else. Never let them get something without making them pay for it. This is a great counter to 5 man caps. Sometimes you need to pick your fights.

Don't chase heroes, especially if you are defending a point.

Rule of thumb, always have more points than them. Kills will usually be about the same for both teams without focusing on killing heroes. Holding points is more important, almost always.

If possible, grab drill/refinery and join top. The reason is right a s yo get to top you level up and get one more skill. (be sure to use movement runes)
Also, if you have someone with ghost or a Rammus on your team that is much faster at the start of the game, Leap Striketo them to move faster. You rarely will run out or mana, so don't worry.

AD - Attack Damage (mostly auto attack)
AP - Ability Power (spells damage)
CC - Crowd Control (stun, silence, blind, fear, knock into air, etc)
lvl - level
Cap(ing) - capturing as in points (windmill, drill, refinery, etc.)
Crit - critical
CD - Cool down (time it takes to cast the same spell again)
hp (HP) - Health points

This is a quick write up for the guide. Hopefully others will recognize the gold within this guide like item order and skill sequence, etc. I will add more to the guide as I notice people have questions/concerns.