Jax is a great character, this build really exploits his opness. The items go in order of what to get and what to build to what, if you have enough money to get the whole item already just buy it and not the stacks. If you're 200g from an item sell you're Doran's Shield and get it, I advise you to do it after level 6, your ult ensures your survivability and with you're Guinsoo's RageBlade you'll survive even without your Doran's Shield, early you'll need it though because Jax attacks really slowly.
First what you need is a Doran's Shield, lane as long as you can.
If you lane well and have enough get a Ninja Tabi get it
If you don't have enough get a Boots of Speed, lane again to get 500g and then get Ninja Tabi
(Jax's stun revolve around dodge, so add Ninja Tabi with your runes and with your Counter Strike to make a high percent of stunning with Jax, even if the other team is full of casters don't get Mercury Treads, Jax's ult gives you more Magic Resistance when activated)
Ninja Tabi is a MUST!
After getting a Ninja Tabi get a Pick Axe
Turn the Pick Axe into a Guinsoo's Rageblade, at this time you should be level 6, above level 6 or close to it
Guinsoo's Rageblade along with Jax's Relentless Assault = super ownage , with this you should be killing the enemy team easily, Jax is best 1 v 1 so try to engage in 1 v 1's
After you get the Rage Blade get another Pick Axe
This time turn the Pick Axe into Bildgewater Cutlass, after this if you have enough automatically buy a Hextech Revolver, and make them into a Hextech GunBlade, if you don't have enough get a Amplifying Tomb and then get the Hextech Revolver, and make it into a Hextech GunBlade
The Hextech GunBlade adds attack damage, AP and life steal
Jax's passive = more AP and AD = more Hp
At this point you can fight 2 v 1's and you'll destroy everyone 1 v 1, unless the other team is already fed
After you get a Hextech GunBlade, you can either go for a Blood Thirster or a Phantom Dancer
Get a Blood Thirster of you're getting into more team fights than not, with a Blood Thirster you get more Hp and Life Steal, if you have both a Blood Thirster and a Hextech GunBlade you can take 2 to 3 at a time (That's what she said XD)
If you're getting into 1 v 1's or 2 v 1's get a Phantom Dancer, with a PD you can get more attack speed,so you steal life faster, PD also adds movement speed with = more chasing ability add that with Jax's Leap Strike and his Counter Strike and you'll own
At this point with either item the game should be over or near to it, unless the other team is good or fed
Now you can have another choice either get a Black Cleaver or get a Banshee's Viel
Black Cleaver adds AD and AS, which adds more life steal and hp
A Banshee's Viel will add more Hp and mana, If you follow my guide every step you're gonna have low mana, this shouldn't matter late game because late game Jax is gonna kill them the second your Leap Strike and Couter Strike hits them along with some basic hits, but if you're killing as much as you should be you're gonna be going back pretty often to get mana early so if you're getting annoyed with low mana get a Banshee's Viel, not only does it add more Hp and Mana but it blocks a negative spell every 45 seconds and it also adds magic resistance
At this point the game has to be over
With this build you can be the best Jax possible,some people try and get a Rylai's Crystal Scepter, which is good don't get me wrong but with my build you add so much AD and Life Steal the Hp you get from the scepter is gonna be less than the Hp you get from your AD, I'd say don't get the Scepter. Although Jax is a great character don't get too greedy Jax can be stopped with exhaust and stuns so know when to run, if you're going into a team fight use your ult always. People that will do well against Jax are; Taric, until level 6 Jax attacks slow so Tarics hardend skin thing is gonna lessen all of Jax's hits, Taric also can stun you and heal himself rendering Jax useless against Taric early, but after you get your Hextech GunBlade you should be able to take care of Taric execpt if Taric is fed already, so try not to feed Taric, if you can try not to lane against Taric. Another person that can beat Jax early is Singed, playing Jax you should try to harass with a Leap Strike + Empower and run combo, but Singed usually can throw you when you hit em, Early when you don't have life steal, hp and damage people that can throw things like Olaf, Mundo, Kennen etc can also annoy you. Jax's weakness can also come from fighting people that can auto stun like Annie and Ryze, early you're not gonna have Magic Resist so people who can stun and cast also can kill you, but if you lane with a caster or a good character you should be fine. Whatever you do don't feed early, Jax can leap to his minions during a run away try leaping to your minions if they're ahead of you. Jax is a great character, but don't get to greedy, don't feed and Jax is better in 1 v 1's than team fights.