Welcome to my 2nd guide with Jax, the Grandmaster at Arms! I'm gonna build him as a AP and a little AD Carry. Lets get started!

Greater Glyph of Potency. Hmm, I take this because of the early AP.
Greater Mark of Strentgh. Good to have this fresh damage to last hit minions.
Greater Mark of Desolation. I prefer this because it got a good amount of armor penetration to win against the enemy carry.
Uuhh, smell it? Oh, it was a dodge. Take a Greater Seal of Evasion for this awesome thing!

I take this because of all the AP and AD, and I chose this under any circumstances!

Doran's shield for the good health and armor. With this you can stay in lane longer.

Mercury's treads, oh yeah. Get that magic resistance. And the best of all you get a Tenacity of 35.

I buy Sheen early because of the AP and mana, and that wonderful passive that when you use a spell your next attack will be equal to your AP. Later I get it to a Trinity Force.

Ok, now the Phage. Always good. Buy a Phage right after the Sheen to get that Health and 18 attack damage. And for not to mention.. You got a 25% chance to reduce you're targets movement speed by 35% for 2.5 seconds.

And then I buy a Hextech Revolver for the good ability power and some awesome spell vamp.

After the Hextech Revolver, go straight to the Trinity Force. And then a Hextech Gunblade

Now buy a Pickaxe, then farm a bit. When you got some money buy a Guinsoo's rageblade.
It gives 35 damage and 45 ability power. This weapon also got a passive. Everytime you basic attack you're attack speed increases by 4% and ability power by 6.
Even if I know you dont get so much money like this, buy an Infinity edge, for the massive attack damage and 25% critical chance!

This is basicly the items for Jax.

Skill choices. Max your W first it gives 40/85/130/175 or what it 180/220 additional magic damage.

Then Q, Leap Strike. You can jump on basicly everything. Wards, minions, enemy and allied champions.

Then your Counter Strike. Gives you a 100% dodge chance for a short duration and after he stun surrounding enemies for 1 seconds. Beautiful, eh?

Teleport. Very good to get back in lane when you've been dead or recalling. It has a long cooldown, but very good.

Exhaust. I've been trying this but wasn't that good. You can catch up the enemy but I would rather prefer Ghost.

Ignite. Alright to have, to ensure a kill or to kill him with out towerdiving

Flash. I take this because it can save your life several times.

Other spells I dont prefer, so I dont wanna write it.

Pros
His Leap Strike can get him away from sticky situations.
He's E, Counter Strike, can be good to fight against enemy AD Carries like Caitlyn, Ashe, Master Yi, Graves and more.
He's W has a very short cooldown and has massive magic damage.

Cons
Don't got a lot of health, so can be easy killed if stunned or snared.
If ganked he dont got suvivability unless Flash and Counter Strike are of cooldown
Easy to get killed if Morgana snares you and her team is coming

Let your tank come come first and then Use Q and then E. If you practised you can use E and use Leap Strike when the stun comes. If you got an Amumu in the fight it will almost be ensured that your team will win

Jax doesn't have alot damage in early game, but the Runes and Masteries should help a little. Try get a fast Mercury's treads and Trinity Force and then go gank other lanes.