Basically, I've mained Jax for one sole reason. He can completely demolish 1v1s and if played smart, even whole teams. Back before he was nerfed to oblivion, people would build him with straight dodge and try to stun spam. Now he is completely different and is required to be played by actually playing with your brain. Most of his skills are based on Damage and AP, which makes him a perfect Hybrid champion, along with his ultimate's passive, he is required to actually attack normally for that burst to pop off. Let's begin shall we?

This is my first attempt at a guide so please don't downvote me without a valid reason.

Being a very bursty and squishy champion. Jax has many Pros and many Cons.

Pros:
Can easily harass early game.
Can deal massive amounts of burst damage and get in the enemy's face.
Can wipe out teams with his arsenal.
Has a good chasing/escaping skill.
Works great in a lane when minions are attacking him.

Cons:
Almost every mage champions can easily counter him by making his only CC useless and quickly bursting him down.
Any DPS hero can easily counter him with just a Sword of Divine.
Squishy late game.
Item dependent.
Needs to dodge to be effective in team fights.

There are many different runes to use with Jax due to his versatility.

I normally go with per level Attack Damage Marks, Flat Dodge Seals, per level AP Glyphs and Flat HP Quints.

With the increase of Attack damage and Ability Power, Jax benefits greatly with his Passive skill. Along with Flat HP Quints to improve early survivability, Jax is pretty sustainable.

You may also switch out those AD Marks for some ArmPen marks, but that'll only work if you are planning on using your Basic Attack a lot since it does not improve any damage for your skills.

As well as Armpen marks, you can go MPen marks. I personally have not tested this so I have little say in the matter. However I think it may do well against teams that love stacking MR.

Switching out AP Glyphs for Flat CDR Glyphs is always a good option. Being able to spam your Leap and Empower makes you a force to be reckoned with early game. However they are pretty pricey.

MR Glyphs are also a smart substitute. If you hate how vulnerable Jax is to casters, you should definitely grab these.

Depending on how you like to test your luck, you can swap out those Flat HP Quints for Dodge Quints. However, I don't like doing this because there's always the chance that there will be a lot of casters, making Dodge pretty useless. In any case, it depends on your taste.

And obviously, keep the Dodge seals. There is no substitute for them unless you want to never use Counter Strike which is your only form of CC.

As you can see, the AD/AP runes DO work for his passive.

Masteries. Now this is also very optimal, but since we're going to go for a heavy Hybrid, we'll go with a pretty standard 21/9/0 build.

Offensive

Deadliness - I personally don't see crit being useful on a Hybrid Jax so just put one point into this just so you have atleast a chance at hitting one.

Archmage's Savvy - I normally put 3 points into this just for the extra AP.

Sorcery - Put 4 into this, you'll need the CDR. Plus you need 4 points in this to get to Archaic Knowledge.

Alacrity - Put 4 points into this just for the IAS so you don't attack hella slow when you're trying to build up your Ultimate's passive.

Archaic Knowledge - Probably the most useful Mastery there is for Mages and Hybrids since 15% MPEN is a **** ton.

Sunder - Put 3 into this so you can smack through some armor.

Burning Embers - Hell, you can even swap it out for Cripple, just get this for an extra point for Havoc.

Brute Force - Put 3 into this, self explanatory, gives flat damage so it'll help your skills a bit.

Havoc - 1 in this is crucial so you can completely smash enemies.

Defensive

Resistance - Put 2 points into this for some MR.

Hardiness - Put 2 points in this just so Resistance doesn't feel left out.

Evasion - An extra 2% dodge, hell with only Tabi's being the only source of dodge, you'll need this.

Nimbleness - It helps in a lane when you are trying to escape or chase. With minions hitting you, you basically have a free Zeal. You could save this last point however, to invest in Haste.

Damn, barely on items. Okay well this is where the fun happens. Jax being an awesome hybrid is pretty item dependent.

First off we go for either a Doran's blade, Doran's Shield, or a Cloth armor and 5 HP Potions.
If you don't feel like having to farm for your money back, just rush the Cloth armor so you can quickly make your Tabi's. If you see that there are a lot of ranged champions or pokers, go for a Doran's Shield for sustainability in your lane. If you suck at last hitting, go for a Doran's blade for that extra 10 damage.

In this case, let's just say you're good at last hitting and you went for the cloth +5 potions.

Okay on your first time back, you should have enough money to get those Tabi's, if not, just buy some more potions and return to the lane and try to get 500 gold so you can afford it.

When you can, try to build a Guinsoo's Rageblade or atleast have the Pick axe and Blasting wand. Once you finish that, aim for either a Hextech Revolver or a Bilgewater Cutlass if kills keep escaping you. Then quickly build a Hextech Gunblade and start ganking more.

Okay here's where you have to adapt to certain circumstances.

If you are being targeted a lot by CC and nukes, grab a Banshee Veil as quickly as possible.

If the other team has less than 3 CC's aim for that Rabadon's Deathcap.

Once you do get that Deathcap, you should be able to easily take down enemy champions with your standard initiation combo, W-Q, then during fights only spam Empower and Counter Strike, then save Leap Strike for chases.

The last items is pretty optional, you can get a Trinity force, Lichbane, or Guardian's angel.

Personally I like to get a T-Force for all of the stats it gives and the 150% damage when you use a spell.

Now if you wanted to go into a more AP orientated build you could go Lichbane.

And if you wanted to try grabbing a Guardian's Angel because you are sick of dying go ahead. My personal experience with it is bad, I normally get killed almost instantly as I'm respawning but I can see it working in some cases.

You can also try getting a Bloodthirster or Infinity Edge for damage. Thirster gives +60 so thats +180 HP from your passive, while Edge gives +75 which is +225 HP. With a fully charged Thirster you could get +300 HP. It depends, if you die a lot, don't get a Thirster.

Out of all of these options I'd go with the Trinity Force. You could also go for a Trinity Force pretty early. Maybe even sooner than the Veil/Deathcap.

UPDATE: I've been trying out Atma's Impaler and I gotta say it actually works really well.

Just look up there on the chart. Typing out a format for the skill seq looks like a pain.

There's also a different way you could build Jax, if you don't want your early game based on Empower, you can max out Leap strike first for survivability/chase purposes. Leap strike's cool down will be drastically lower but your early damage out put will be as well. Maxing empower first allows you to constantly harass your opponent with huge bursts.

First off just take Leap strike so if you get in a sticky situation you can just leap strike to a minion to escape. Or to get that last hit on an enemy champion early.

After that get Empower. Once you get this pretty high, you can dominate your lane pretty well. Just use it then leap strike and it'll be a hella high burst.

Get at least one rank of Counter Strike before 6 so you can stun and dodge.

At level 6 get your Ultimate. When you get this, you can kill nearly everything with basic attacks. You could even start jungling pretty well, but Jax is much more useful in a lane.

After 6 you can either max out Empower or Leap Strike. Get Empower if you want to do huge bursts or get Leap Strike if you want to be able to chase more.

I normally take Ignite and Ghostwalk for my summoner spells.

Ignite - For those last couple of drops of HP or for champions who love healing.

Ghostwalk - With this you can chase/escape. I usually use it to chase or initiate, use this wisely because if you are all alone without it, you're guaranteed to die.

OPTIONAL SUMMONER SPELLS

Exhaust - In those 1v1 fights, this will make Jax unbeatable. Also used well for those champions who do massive channel damage, such as Fiddles, Katarina, Nunu, etc.

Flash - I don't know why you would need this but I see some Jax player with it. With leap strike you already have a sort of flash spell. Useful in jungle though to go through walls.

Cleanse - Grab this to wash off any CC that may come on you. Normally there's atleast 2 annoying CC's that you want to watch out for. Don't blow this on a stupid slow unless you are right near death.

Well, obviously with the low armor and MR, you don't want to be the one taking all the heat. Make sure you have your tanks do that then come in an clean up.

When you are laning with a support champion like Soraka or Sona, make sure you make the most out of their heals by constantly harassing the opponent.

Basically, Hybrid Jax is a very versatile build which you can dominate with. Always remember to stay in the back and wait for the perfect chance to strike. Jax should always be played as an Assassin.

Yeah so this was my first attempt at making a guide, I hope to get some tips. Sorry I couldn't get any flashy images. Hope you guys liked it and make sure you give some feed back.