Ok, this is my first build hope you enjoy ^.^
I have played Jax many times and this build its pretty interesting, you can kill all in 1vs1 with this build I think. Basically this is a build for 1vs1 Jax and push very fast.

Enjoy ^.^

I like to go offensive, so I can push and kill all. I know that people like to go defensive but i prefer offensive. The offensive masteries help you farming minions to give you money and you stay feed >=), and it gives you attack speed and damage to go 1vs1.

Mark of Desolation- I like this rune because it gives you more armor penetration and that its equal to more damage. =D

Seal of Invasion- I choose this one because it gives you dodge and with that you can use your active from the counter strike.

Glyph of Alacrity- This is GREAT, attack speed is great on Jax, late game with the items you just pown all =P

OK, I like to start with Boots of Speed and potions to stay in lane the much time I can, if you had killed already you should buy Bilgewater Cutlass, if you dont buy Vampiric Scepter, and then finish the Bilgewater Cutlass. After buy Guinsoo's Rageblade and you can kill almost everything in 1vs1...

Ninja Tabi- It gives you armor and dodge just... no words...

Guinsoo's Rageblade- This is an awesome item for Jax in my opinion, it gives you damage,attack speed and ability power, its pretty bad *** when its together with your ultimate passive

Trinity Force- Awesome item in 1vs1 together with your attack speed it just do so much damage =o

Phantom Dancer- Attack speed and crits +.+, and movement speed (very important, so they can run away)

Hextech Gunblade- Its another awesome item on Jax, it gives very damage on abilitys and on melees and its gives you VERY survivability AWESOME +.+

Frozen Mallet- It makes you more tanky and more hard to take down its cool ;D

Flash- It takes you out of some bad places, its also good to chase someone.

Exhaust- Pretty good on Jax, you dont have any slow so this should stop them from run from you ;D

OK, harass them with w+q, and when they are with low healh you whait untill you dodge something (to use the active from your E) you put on W jump to them with Q use stun with your E and go after them if they run >=)

IMPORTANT: When you have Bilgewater Cutlass and Blasting Wand you should go DRAGON!!! You can kill it easily and it give money to your team, and that help you to win the match.

Red Buff: If they are running out from you and you are tired from it, just grab red buff it slow them and they won't run from you anymore ^^

Hope you enjoy, sorry for my bad english and PLEASE say something that I wrote bad if you find or something like that =s

HAVE FUN!!