Hey guys, i'm here posting my first build for Jax. I start to lay with him and he is a very good Champ (alittle OP ;D), and i want to show you what i do for him and maybe you guys can help me telling me with i'm wrong or what.
So get your popcorn, sit back and go to it.

Ps: I'm sorry about my English... i'm brazilian and i speak better then i write. ^^'

Well... i think all of you make this rune tree and somethink i see looking to Jax builds is: They make a Lot of AP but no Magic Penetration or something that diminishes their Magic Resist.
So i get some Magic Penetration from Runes and somethink to help on items.
The runes are very simple... evasion, magic pen. and AP. Not to much, just the usuall.

Now the masteries... well the masteries too is very simple, like i said magic pen is a good for Jax cuz almost all his atacks is magical and i got Exhaust as mastery, well i don't know what u guys are thinking but i got it cuz like i said Jax is almost a Mage with his magical atacks, so Exhaust Mastery remove some Magic Resist and Armor, it's not too much i know but it's a good way to kill the enemy faster.
Then i make a Def build... well you need to kill the enemy first with you want to live forever and we canhave more dodge.

Now let's go to the important part, the items. I start with Doran's Shield to keep in the lane almost forever, if you see you lose too much life keep on the bushes or if you almost die back to base, don't be a crazy man jumping in every champ you see, wait the opportunity and then gethim.
My items is a little diferent of you guys do, i don't get tons and tons of AP... cuz to me a good Jax is an Immortal Jax and to make it i need health, and to get health i need AD... that's why i make that Atma's Impaler (that something i can change by a Sword of the Occult), so as i said Jax is a Magic Damager i got the Abissal Scepter that reduce the Magic Resist from the Nearly enemies and help you kill them faster then normal especially tanks. Two items that Jax can't live without is Hextech Gunblade and Guinsoo's Rageblade, these items save my life a fell times and something you guys need to do is use the Gunblade it help alot! Rylai or Mallet, there a question i make myself for a while, and i see Rilay give's you AP to all your abilities but not too much life as Mallet, that something we need to think Health or AP?
Some good items if you are dieing too much for the mages like Veigar and his Massive Damage you can get a Hexdrinker, it'll help you alot, that save me alot of times! xD

Start with Leap, if you have a good line partner you can get the first blood with leap and exhaust, but all depends the enemy mastery and champ. Second you go to Empower and Third Counter-Strike. Empower is the first skill you need to charge, you'll kill the enemies faster with it. Then go for Counter-Strike, you'll need to get it so you can Dodge and stay closer the enemy for a long time if you want to kill him. ;D
Get your ult always you can do it, and remember to use it when you gonna die for a magic damage!
Boost the leap after all, you don't start a fight NEVER! Let the tank start then when he get focused start killing the DSP/Mage/Stuner. Jump with the Empower to get more damage.
In a 5v5 fight try to use your Couter-strike every time you can do it.
If you are going to die, stay and try to get life with life steal or jump off in a friend champ or creep.
Try to use Empower every time you get in a fight, it'll heal you alot.

I choose Exhaust cuz like i said before it reduce the champ MR and Armor for a time and it help you kill him faster and i use Cleanse cuz... well... everybody hates when you use Jax and they have a lot of stuners or slowers! ^^
But you can change Exhaust (cuz you already have a Slow [Rilay]) for Ignite to kill runners, Flash to get closer when leap is on CD or heal with you are insecure when you jump to the fight.
The Cleanse i think is the best choose but if you want to change, you can get Ghost to run faster or Teleport to get ready to the fight faster.

Well this is my build please comment and vote, i make this build for a long time and i always get 15 + Kills and 9 - Deaths and 9 + Assists, i'm sorry don't put here the print screen to show you but in other time when i make another build i'll put it here.

Thank's for comming.
GL HF