This is my first Build here I hope for any critics. (I apologize right now for my poor english)

The purpose of this build is an AD Jax you can play solo top (I prefer this way) or dps bot.
Jax can also be used as a Jungler but I won't speak about it in this Topic.

Jax is an assassin/Melee champion. With this build indeed he is both.

The difficulty avout Jax is his early game. He remains quite weak although last patch has up his stats. All the difficulty is here because if you manage your early Jax will quickly become unstoppable.

Why an AD Jax? And why this Build?

He has less burst than AP or Hybrid but he makes more durables dommages. Notice, empower is the only Ability power dependant skill of Jax. Moreover this build is really balanced between the damages and the survive-ability.
Opponents Team will have no good choice: They can focus on you but you are a HP bag with a good defensive ability (stuff+ulti). They can Ignore you but you are a huge damage dealer able to focus easily any of them with your jump.... As you see no good choice for your opponents.

Greater Mark of Attack Speed I chose speed attack instead of attack ou armor penetration for the marks. I prefer those because of the passive bonus attack of the ulti which is easier to obtain.
Greater Glyph of Magic Resist and Greater Seal of Armor give you the Armor and Magic Res necessary to enforce your squishy Jax in the early game.
Greater Quintessence of Life Steal Steal of life for the quintescence with the Mastery it gives you 9%. You will need themto maintain health and hold your line.

Mastery:

21/9 is for me the best choice. 9 points in the defense tree are necessary for the early game. You can modify the ratio Resistance / Hardiness according to your opponent.
As for me, the only choice in the first tree is between Sorcery(4)+ Arcane Knowledge (1) and Havoc (3)+ Blast or Mental Force (2). Both options are viable. Mental Force (4) is a possibility too but seems less attractive to me.

The Spells:

Exhaust and Ghost are my first choice because it's an incredible combo for ganking and exception made of jungle fight ghost is almost good as Flash to escape quickly and really better than it to purchase ennemies. Moreover you can improve both with just one Mastery in your main tree...
Of course Flash is a good choice too.
Some peoplethink Cleanse is necessary to play Jax because he is too weak against CC (which is really true)... I don't think so and I will explain why later but it's a good choice too.

I will especially focus on the early game and early choices because the rest goes without saying.

I tried all kind of items to begin the party: Doran's Blade, Doran's Shield, Ruby Crystal, Regrowth Pendant. All of these have favors and inconveniences. I look especially if the item can be upgraded later because Jax's Stuff spends a lot and 244 gold (doran bought-doran sold) can make a real difference in mid game.

Doran's Shield is probably the item which give Jax the best survive ability early game especially against AD champions. The inconveniences are the fact that you cannot use it to build better items and that it's of course not the best item to obtain early kills. Against an AD range you will hold the line but you wont afraid him => he will dominate. However against an AD enemy you have no chance to kill 1 vs 1 (for exemple some off tanks), this item may be the best choice.

With Ruby Crystal and Regrowth Pendant You have items you can upgrade later but with the first you have mass health but no regeneration and the opposite with the second one. With both you don't afraid your opponent at all => he will dominate you.

Doran's Blade is good item for jax but because of the multiple nerfs of the life steal i don't think it's the best choice. 12% is better than 9% but as for me it's not enough...

So my choice is Boots with 3 Health Potion. It gives you movement speed which is the only way to afraid a caster on your line and the potions + passive life steal will give you a good durability. You can early kill especially bot with a good ally lvl 3 you let them push focus the more vulnerable Empower+Jump+stun+Exhaust+Ghost=kill.

The great question what should I buy then? If you are against and AP no question, you buy Null-Magic Mantle to end Mercury's Treads. If you managed an early Gank you can buy Sheen which will allow you a better burst (for an other gank) and will end mana problems.

Once you get both Sheen and Mercury's you will need Zeal to increase speed and especially movement always in the same objective: kill heroes and farm minions. Then you will finish Trinity Force.

Why I chose trinity is because it is a really complete Item. It gives you all you need not less not much in a perfect synergy. I won't repeat all its skills. As for me Trinity is way better than Hextech Gunblade which doesn't give any speed movement/mana/life bonus, the slow bonus is active (and I'm Idler so...) and more important trinity under Items are way usefull in early than hextech one's.
I can say you that Jax is not the same with or without trinity (even Phage+Sheen+Zeal vs Trinity) which is not really true with the gunblade. And if you get it sooner enough with a good early you made half the way to the win. Now opponents will have huge difficulties to escape from your ganks.

Next step is to complete Warmogs Armor ((start with the belt then the ruby if you can). This Item will gives Jax his survive ability. Ennemies will hesistate to focus you when they will see you at 2600-2800 HP lvl 12 and almost 3400 HP lvl 18.

At this point you have a quite good damage ability, a lot of HP, a correct magic res. You will need armor => Chain Vest will be helpfull and if you upgrade it into Atma's Impaler. You've done the Job cause ATMA will give you 45 armor, 18% crit AND about 60 AD. You have now about 250 AD not too bad!

With these 4 Items, your Jax is complete, other Items are just bonus!

After the skills you may want (cause we always want more) are magic res which starts to be low against casters, attack speed, some steal of life always interesting and more AD and AP if possible.
If we dominate, my preference goes to Maw of Malmortius especially against a mass AP team. It gives you more AD, a bonus in magic res to equalize your armor score and an interesting shield ability when low HP (which can give you the one or 2 secs you need to reburst some abilities like your stun or permit you to escape).
If the adverse team has mass AP and mass CC, Quicksilver Sash is a good choice and cost less gold for a better magic res. Even though its mana boost is obvious, Banshee's Veil can be a choice it gives you 375 extra pv and 7 AD of course but I think Sash is better and way cheaper.
Increasing attack speed is also a good choice with Guinsoo's Rageblade or with a Recurve Bow that you can turn in Madred's Bloodrazor or Ionic Spark (health bonus = more damages). Zeke's Herald with it HP bonus, life steal and attack speed bonus is a very good choice too I take it as 6th Item if the party keeps going.
Of course Hextech Gunblade is also a good choice.

You have understood, the aim of this stuff is to build a tanky AD carry who can survive in a team melee while dealing massive damages. A little bit in the spirit of the old Jax when his passive ability allowed it (way better I agree).

Let's come back a little, you were gone for an easy early kill and.... EPIC FAIL! You die, feed your opponent and mortgage the rest of your party. Now be sure that if you are solo top you will undergo for the rest of the line fight unless the jungler helps you to invert the vapor. With Jax you have few rights to mistake early game. If it occurs, undergo, tighten your ***, stick to your tower and survive. No feeling with the tower if the jungler comes to help your opponent => run, and call for help. You also might have to change the order of your Item buy. The upgraded shoes will be needed (to be able to escape) immediately and then Phage will help you with its health bonus and slow down (helpfull to avoid a gank or at least die with your opponent under your tower for exemple).

Empower is for me the first skill to get cause it will be helpfull to creep minions Jump is obvious lvl 1 be sure you wont gank anyone. Moreover the time your opponent see you can't jump, you'll be lvl 2!
Next I let you the choice. Always in the same mood, I tend to choose Counter Strike which can avoid many damages in some cases and allows you to be ready for lvl 3 and why not a gank try. Once more your opponent doesn't know your choice and he can think you are able to jump which will make him play carefully and loose some creeps.
lvl 3, complete with your third skill. Now you are able to do something against your opponent.

Then I advise to boost Empower. Mana cost don't move and CD decreases. It's the way which allows you the best creeping ability and the greater dps.
Once more, the choice to upgrade the jump or counter or both can be discussed. Personnaly I upgrade first counter because when his CD decreases at the jump lvl=> for a purchase (you get it back at the same time than the jump => Jump=> Stun> Kill. Moreover, this ability is the more helpfull during a team fight.

With Jax until lvl 6 creeping is not simple you will have to use mass empower cause you don't deal mass damages => Low mana; while you will have to check for a gank or an escape.
Against Range Heroes, you might be in trouble in particular AP range heroes (against AD you have counter strike to help). Take care until lvl 3 because if you get too much damages early you wont be able to afraid him with the potency of a gank.

Real problem for jax are Heroes with early CC like Ryze, Annie, Riven... They will make you suffer and getting initiative will be hard. For example ryze stun you from distance and attacks you, riven charge you=>stun=> you cannot use Counter strike until you are mid life...
You will tell me of course that's why cleanse is necessary to Jax. Yes an No. Cleanse CD= 3-4min. CC of these heroes 15 sec to 60 sec.... I let you make the count.

Early game will be difficult and you will need your jungler to help you. However, you need only one kill and the free creep time you will earn to invert the vapor. The rule is simple play very safe, stay alive. Even though you miss a lot of creeps. You've better be alive and ready for a kill (which will afraid your opponent) than creep some minions more and be low life for a while or having to back continiously and letting the line free to your opponent!

Team Fight:
Warmogs Armor need you to creep to upgrade it. But, don't do it, never, to the detriment of your team. When Team movements start you are with your team! (Notice a kill or an assistance will up warmog very fast! A victorious team fight is way better than 15-20 minions/monsters creeped at the opposite of the map)

Mid Game you are not complete and so you are not very tanky and just a correct damage dealer (of course if you are fed compared to ennemies you are both....). Don't engage the Fight!!! Never! Let the tank engage, and then enter the melee. You can easily focus the targeted hero with your jump. Just before activate Counter Strike and Grandmaster-At-Arms. They allow you a great capacity to endure damages. Don't hesistate to activate the stun early especially if there is a great concentration of ennemies or if your team is loosing! This second might give your team the time to kill the first opponent and open the way to victory!
Now people can see good Jax players by the way they use Counter Strike +> get the good timing!

End Game You are now a great damage dealer with a great survivor-ability, you chained killed, you are fed, you feel invincible that's why you can now engage the fight as you want...

NOOOOOOOOOOOOOOOOO You've been blinded the Jax's invincible feelling! You're done! You feel invicible and start trying incredible kills => ERROR! You are not a tank and neither your stuff nor your ulti wont protect you against an early focus or against your stupidity! You dead, your team losts its carry AD and and is engaged (because she wanted to save/follow you) => Best way to a 5-0!
The tactic is the same as previously let the tank engage, even let a little latence before jumping in the melee each CC already casted is a CC of which you won't be the target.
YOU ARE NOT INVULNERABL fight against this feeling! Most important your team need you alive in early game the team has carried you and at the end it's now your job to carry your team!

Should I insult readers saying who to focus or not focus? I will because experience of incredible Team Fail let me on the *** more than once.
-Don't focus the Tank! Never! Even if he is mid life!
-Look the stuff of your ennemies there is almost each time a kikooo stuffing all attack, no defense (often Caitlyn or Master Yi players). They deal massives damages but Jax will down them in 2-3 secs...

If you are here, you have probably read all of this topic thank you for your patience and your tolerance.

I'm hoping for your critics and propositions.