Hello there, I'm Bryan.
Browsing around MobaFire I looked at the guides for Jax and couldn't really find any I liked at all. So I would just like to share a build that I think works REALLY well with Jax and is created by a top tier ELO player who actually plays Jax a lot!

Original Guide - Dyrus's Guide.
(All credits go to this guy!)

It is seldom the case where a champion is defined by his actions after joining the League of Legends rather than before. Such is the case with Jax, for whom the argument could be made that he is the most prolific tournament fighter currently at the Institute of War. Before joining the League, Jax was an unremarkable soldier-for-hire. For reasons known only to the former leader of the League, High Councilor Reginald Ashram, Jax was put on the top of the list of candidates to receive a League Judgment - the interview process that either accepts or rejects a prospective champion. His Judgment was the quickest in League history, where the Doors of Acceptance glowed and slowly swung open as soon as it began. Jax faced no recorded Observation or Reflection during his Judgment.

Jax proved himself to be an immediate terror in the Fields of Justice. The self-proclaimed ''Armsmaster of the League'' rattled off a streak of consecutive wins that to this day has not been matched. A number of summoners in the League grew concerned that the perceived objectivity of the League of Legends would be questioned by the presence of an unknown fighter who was unbeatable. For this reason, the new leader of the League (following Reginald Ashram's disappearance), High Councilor Heyward Relivash, created special restrictions for Jax to fight under. This was something the League had never done before, and something that has never been done since. The burly fighter responded by imposing his own special conditions; as a means of protest, he permitted himself to fight using only a brass lamppost. Neither the League's sanctions nor his own has affected his winning ways. The League has since rescinded its sanctions, but Jax has not; he fights and fights well with his trusty brass lamppost.

''Be advised - there has been an outbreak of lamppost-shaped bruises in the League of Legends.'' - Gragas

• Relentless Assault - Increases Health with Ability Power, and Attack Damage. This is why you get 3 Greater Quintessence of Attack Damage, & 9 Greater Mark of Attack Damage along with Doran's Blade. They increase your health significantly for early game so you won't be a squishy. (That is why people complain he is under-powered, because his early game is terrible, this fixes it.)
  
  
• Leap Strike - A skill that inflicts damage by jumping towards target. Can be use to chase down a target or escape. Always get this skill before Empower you may think Empower does more damage, but at level one Leap Strike actually does, and gives you better mobility.
  
  
• Empower - A skill that increases Jax's damage. This is what makes Jax so powerful, with this move along with his Leap Strike and Relentless Assault passive you'll be taking down squishies so fast.
  (Tip) - Use Leap Strike THEN Empower, not Empower then Leap Strike.
  
  
• Counter Strike - When Jax dodges an attack he is able to use this skill, an AOE stun that lasts for a second each time it's used. Always try to harass as much as you can when this ability becomes active. It gives you time to Leap Strike in Empower, stun then get out.
  
  
• Relentless Assault - Jax's Ultimate, it's passive increases Attack Speed, gives bonus magic damage, increases dodge chance, and increases Magic Resist. Always use this when you are about to enter a team fight or if you are dueling against a caster. Jax's base magic resist is low so this would help a lot.
  

• Greater Quintessence of Attack Damage - Increases the health on Jax due to Relentless Assault and the flat attack damage runes scale with Leap Strike and Empower to dominate lane early game, you will be taking the squishy health by half.
  
  
• Greater Glyph of Magic Resist - Magic Resist for extra survivability.
  
  
• Greater Seal of Evasion - Increases dodge chance for Counter Strike along with Ninja Tabi
  
  
• Greater Mark of Attack Damage - Same explanation as Greater Quintessence of Attack Damage.

Main Spells -

• Ignite - Damage over time spell, useful for first blood, or destroying the last needed health on your enemy to receive a kill.
  
  
• Flash - Used to gain distance between enemies. You can also use Flash to escape any gank or team fight if it isn't favorable to you. Flashing through walls can save your life and take one. Leap Strike+ Flash = Amazing, similar to Akali's Shadow Dance+ Flash combo.

Other Spells -

• Exhaust - This spell can decrease the speed and attack of the targetted enemy champion, this can be used in many ways, such as to catch up with Leap Strike, to escape, or to finish off an opponent if you're low on health.
  
  
• Ghost - Allows you to gain movement speed to chase or escape from an enemy champion effectively. Able to run through those pesky minions that get in your way!
  
  
• Teleport - This spell is very useful to protect your turret(s) or even an ally(s) when needed, blinking in by teleporting to a minion or ward in the middle of a team fight can change the tide of a game.

Masteries -

Offense - 21 points allocated in here to dominate Early game.
Defense - 9 points allocated in here to increase dodge chance, and magic resist.

• Harass with the Leap Strike, Empower, Counter Strike combo.
• If the opponent is low enough get First Blood.
• Don't be afraid to use your mana if it's full.
• Zone out your enemy using brush.
• Deny enemy of experience and farm.
• Your first trip back should get 2 more Doran's Blades or a Bilgewater Cutlass.
• Farm

• You should have Hextech Gunblade.
• Start dueling enemy champions you are basically god.
• Start Sight Warding key places for Map Awareness.
• Keep a close watch on Dragon and Baron ( Sight Wards).
• Jax is an extremely good tower pusher so any chance you get take that tower.
• Make sure to get the the Red Lizard Buff it's virtually impossible to get away from Jax with it.
• Farm

• Should have Atma's Impaler, Banshee's Veil, or any other items you have chosen.
• Map Awareness with Sight Wards.
• Should be taking out the carries.
• Continue to split push or even backdoor if possible.
• Farm

• Come in later after team fight has already initiated, and go straight for the squishies.
• Counter Strike as much as possible.
• Don't forget to use Relentless Assault.
  

Tips & Tricks -

• Use Elixirs when needed (ie. When they have baron buff, or you are behind.)
• When your Counter Strike activates at any time try to harass or even kill your enemy.
• Sight Ward key places when you are soloing or just for map awareness.
• Farm as much as possible
• Always get the Red Buff or Blue Buff when it is up.
• If you are losing, try back-dooring. Jax can take down a tower within seconds.
• If you see anyone pushing a lane alone, make sure to punish them.

• Doran's Blade

• Cloth Armor
• 5 Health Potions

• Boots
• 3 Health Potions

In the item build above it says to get 2 more Doran's Blades. But, if you have enough money go straight to Bilgewater Cutlass and don't bother buying the other 2 Doran's Blades and just go ahead with the item build.

• Ninja Tabi - For the extra dodge chance for Counter Strike.
  
  
• Hextech Gunblade - Extra damage for Jax's auto-attacks, Leap Strike, and Empower. The ability power and attack damage and other specs is just perfect for a hybrid like Jax. This skill also has an active passive that is much like an Ignite with a slow. Place this item in the first slot by dragging there, and to activate it press 1 and click the target. Remember to use it!
  
  
• Guinsoo's Rageblade - Increases attack speed, and damage for Jax. This item synergizes with Jax's ultimate so getting to 8 stacks should be a cinch. This item was practically made for Jax, literally. Guinsoo played against Dyrus's Jax and specifically made this item because of him.

Damage -

• Doran's Blade - Increases Jax's health and damage. With Relentless Assault this is just what Jax needs to survive early game. Stacking these is ideal.
  
  
• Rabadon's Deathcap - Increases the burst damage for Jax's abilities. This item is a must since Jax's abilities are ability power you need to have a mix of sustained damage with the burst of his abilities, and this item gives it to you.
  
  
• Sheen - This item creates a huge amount of burst early/mid game or even until late game comes around, the way it works is when you use an certain ability it will activate dealing your next attack 100% extra base damage.
  
  
• Trinity Force - Increases damage. If you are totally fed and ran out of items get this. It has basically every stat Jax needs and a slow too.
  
  
• Void Staff - Used for a heavy magic resist composition, and increases burst damage. If the other team has a high magic resist just grab this, and it increases your ability power.
  
  
• Bloodthirster - Gives life steal, and gives Jax a sustained damage. This will also give Jax more survivability, just attack tons of minions and you'll see your health bar increasing in no time.
  
  
• Black Cleaver - Increases attack speed and damage if you want Jax to hit faster.

Sustainability -

• Banshee's Veil - Increases survivability and blocks one spell. Use this when there are pesky casters on the enemy team such as Lux, Ezreal, Veigar, Malzahar, etc.
  
  
• Atma's Impaler - Increases armor, and damage. This provides Jax with the sustainability in team fights and gives Jax more sustained damage.
  
  
• Guardian Angel - Increases survivability, and gives Jax another life with 750 health. I love this item so much, who doesn't want another life. This guarantees a kill or even 5 of them. ;)
  
  
• Warmog's Armor - This item is to increase Jax's health, if there are no casters or there is no reason to get Banshee's Veil you could go Atma's Impaler and Warmog's Armor for more health.

Countering -

• Quicksilver Sash - Great for when there's tons of crowd control on the enemy team, and to counter ultimates like Infinite Duress by Warwick, Nether Grasp from Malzahar, etc.
  
  
• Wit's End - Increases magic resist and attack speed to counter casters and hit faster on Jax. This has a stacking sort of passive that increases as you attack so make sure to have this up before you enter a fight against the caster, because they can most likely use crowd control on you where you can't even hit them.
  
  
• Sword of the Divine - Buy this when you want to counter the enemy Jax. This isn't that recommended but if the other Jax is completely fed and you happen to duel him a lot, just grab this, it'll give him no dodge chances from you.

Update - Patch v1.0.0.121




Yes! This 'skill' is implemented back into the game, where Jax can Leap Strike onto a Sight Ward. There are various of ways to use this to it's fullest potential such as gaining ground or even escaping to places where you would need Flash to get too.

Here are some examples below.



First off, this is the range of your Leap Strike compared to the range of a Sight Ward so you shouldn't have any problems using Leap Strike on the Sight Wards since Leap Strike's range is larger.



Key places to place Sight Wards are shown above, since the map in League of Legends is symmetrical you can place Sight Wards on each side of the map with no problem.
Ridges - With many ridges in the game you can use them to your advantage by using Leap Strike on a Sight Ward as shown above. This technique can save your life or even take one.
Baron Nashor & Dragon - This is one of the most useful places to place a Sight Ward down where it's a clear cut escape, unless they have Flash.
Towers - I have not yet provided a picture to this but you can also place a Sight Ward in front of the tower you are laning in to seek safety faster in case you are being ganked or simply losing against your enemy.

How Dyrus plays Jax.
Credits to Dyrus.

I hope this guide has helped you become a better Jax player!
Make sure to read, critique, and enjoy!