Second build and one of my ways of building jax, though you never know how you should exactly build a jax, so you may have to improvise a few of the times.

Empower -> Leap Strike (stun with counter strike if its useable) and then keep on attacking and using empower whenever its off CD. Also, take advantage of being near a minion wave, since if you dodge one of their attacks, you will be able to stun the player attacking your champion.

Leap after any running away champion, but DON'T CHASE unless you know you will get the kill and will get away without dieing, you are a carry, you DO NOT want to die.

Last Whisper: Its the best item against a team of tanks.

Executioners Calling: Great against a healer team or a mundo and vlad.

Lich Bane: Great when your versing a high magic damage team.

Sword of Divine: If your completely wrecking the other team you can get this for more DPS

Mejai's Soulstealer: Only if your wrecking the team and want to deal more burst damage.

Ghostblade: If their building a lot of magic resist then you can get this for more AD

Thornmail: Fighting a high AD team and your getting focused down by them a lot.

Ice Mallet: If people are getting away from you a lot or need a little bit of survivability.

Sword Of Divine: Versing a teemo (blind) or another Jax (Dodge).

Well...theres tons of other items you can get but these are the most likely you will get.

Expected to...

Deal High Damage (OH YOU BETTER BE DOING HIGH DAMAGE!!)
Assassin weak enemys while their not expecting it.
Be able to harass with Empower + Leap Strike
Carry...well maybe I usually carry as jax, but everyone has their own cup of tea.

What your not expected to do

Tank, if you try to initiate a fight as jax you better be uber fed or drunk. Because if you come back here and complain about dieing because you thought you could tank, I WILL ADD A PERSONAL CHAPTER FOR TROLL COMMENTS JUST FOR YOU!
Jungle, Well you can but usually only late game.

Cant think of anything else, jax fills out a lot of roles..

PROS

~ Your pretty strong levels 2+ due to how much damage Empower + Leap does.
~ Fills a lot of roles in groups so you can almost always use him in matches.
~ Once he is fed hes unstopable

Cons
~ Targeted a lot
~ A planned gank can ruin your day
~ Your a guy wearing purple....shall I say more?
~ You use a lampost as a weapon.

Hit the minions when their hp is low enough.
Farming gets a lot faster once you have your ultimate..

Yea...can't think of anything else to put here.

Dorans Blade ~ Gives damage and lifesteal with some HP for survivability in early game.
Ninja Tabi ~ Dodge, jax..do I really need to explain this..Can switch for Mercury Treads if they don't have many melee.
Sheen ~ Will make your empower + Leap strike combo even more powerful.
Hextech Revolver ~ Gives some AP and lifesteal after using spells, Idk much about this I just got it and it stuck.
Rageblade ~ Works very well with jax, gives ap, ad and a Ult - like passive.
Hextech Gunblade ~ Gives both types of damage, a slow with some damage and some lifesteal.
Trinity Force ~ Gives ALL good stats for jax.
Blood Thirster ~ A good AD boost with some lifesteal
Deathcap ~ Properly named gives a lot of AP, sell Dorans Blade for this.

Yea....

Thanks for reading my jax build.