This is my first build so constructive critiscm is wanted (:

Ok jax is one of the most BA champs in LoL, he is simple to play and will produce a lot of good scores. When playing jax you have to know that you are not a tank, but can take a little bit of damage - you also need to know when you can win the fight and when to wait for back up. To play jax you will need a good knowledge of LoL but you won't need any amazing last hitting skills or fancy footwork, however to truly master his powers you will need a lot of PRACTICE and a lot of changing things up and trying your own playstyle.

I have turned off require comment to vote so feel free to be anon

Pros
-Great fun to play
-Does a ton of damage easily
-He is one of the easier characters
-His Passive will allow you to not die too much
-He carries a ****ing staff? thats BA.

Cons
-He gets targeted at early game because he is a very late and mid game op character
-You aren't the best at initiating
-You cannot get that many multi kills when going rambo style (1v2/3/4/5/) like people like irelia can if there is a lot of CC, if not go for your life.
-For me personally if I don't get a good ish early game I don't get a great late game (not to say he can come back very very well)

Ok Runes do play a big part in building Jax, I really really really really recommend getting the dodge seals, without them you will not have enough dodge to be super OP - as for the others the Marks can be either Magic penetration or Armour penetration, I prefer the Arm Pen simply because I believe your spells are already extremely good but when your cd's are done it's sometimes hard to get that last shot in, with the marks your attacks will hit just that bit harder. As for the glyphs it doesn't really matter wheter you want to go with full AP glyphs, I just like to have a bit more CD reduc and you may want to try it.
Dodge quints are also very important - It will boost your dodge up a lot and when at lvl 30 following the build shown you should have a sufficient amount of dodge to get your e active 3 seconds ish (:

There are a few summoner spells that would be alright and some that really DO NOT WORK for Jax.

My Choice in spells:

Flash: It's simply a great spell and will allow you to get your chase on and get those stuns in range.

Ignite: Helps you with your damage but if you want to go more defensive get one of the others.

Things that work:

Exhaust: Exhaust is a great summoner spell and it's pretty simple why, it will give u an early game advantage and will allow you to get some easy kills.

Ghost: Another great spell and will work well with Jax when you need to chase, also if you need to run away it will help.
If you are not completely comfortable playing Jax - this spell will help you stay alive a lot.

To be honest I wouldn't go with any others maybe, maybe maybe cleanse? it will help you a lot at certain points but if there isn't much CC on the other team it's preetty shiii'teee.

Edit:
I have heard that cleanse is a good spell for Jax, so I tried it out and it does work quite well , just make sure you use it at the right time and not when you are getting cc'ed by one person. If you use it in team fights you will have a good edge on those characters which can just run away from you after using slows/stuns.

A Brief Overview of the your Spells

Equipment Mastery
Your passive gives you health every time you buy an item no matter what stats it gives - giving you a very off-tanky character (:

Leap Strike
Leap strike is a very powerful spell, it basically allows you to jump to a target and if it is enemy it does damage - you may also jump to a friendly target (towers do not count ;) ). It is very useful for running away from enemies and ganking from the brush.

Empower
This is along with Leap Strike is a ****ing amazing combo - pretty much you will want to enable your empower and then use leap strike. It does a ton of damage and will destroy squishies early game. So empower what it does is it scales with your AP and AD and will make your next attack to a ton of damage - it also resets your auto attack.

Counter Strike
This is soooo amaazing this spell is soo amazing liaefjoisjfoaijsfoijasf. It really is, firstly it increases your dodge chance, then when you dodge an attack you have a chance to stun someone.
You will have to get use to the timing of it, but how I use it is I get it up by attack minions/jungle creatures and then jump in and stun straight, you will then want to stun again at around 1/4 health making sure that they stay close when your going for the kill, it also makes sure they don't run away too far when your about to kill them - so you can leap strike to them.

Relentless Assault
The passive of this spell is great at low game for farming minions and then later on it is beastly when pushing towers and going in for ganks. Also it is great for defensive times when you are going against AP characters, basically it increases your magic resist (:

Ok so the items shown up the top are my basic late game 6, if I get the chance I sometimes change the Ninja Tabi to a phantom dancer for the extra crit but to be honest that doesn't happen that often. The Gunblade and rageblade are MUST HAVE ITEMS! seriously you must buy them, and some people decide to get sheen before finishing the gunblade i do not because I don't buy a lich bane (i know its strange) but yeh thats how I do it.

When buying your items if you have the choice of buying a health item before an ap/ad item ALWAYS get the ap/ad item - you will still get some health when buying the item and get the ap/ad where as if you buy the health item then you won't get the ap/ad.

Early game you will want to buy Either:

If you are new to jax / lol

Dorans Ring then go onto buying boots and gunblde and then onto your rageblade

If you are comfortable with jax I would recommend buying boots and three health pots, this is because the ring is usually used to stay in lane longer and you will not need the ring late game so it is easier / faster to just get your boots and be able to run away.

However, if you do buy boots you will need to not harras too much or you will go OOM, just choose your opportunities and don't jump in at every opportunity.

Ok I have skipped Mid game simply because there isn't much else to say, just keep building towards your final set and try to gank as much as possible - I would recommend group ganking in early game as you aren't very strong then but late and mid game you will be able to kill many many many people.
Ok late game what you will want to do is try to get the people who are trying to push towers, jax is very good at fast burst and chasing up to enimies, the scepter will help you to chase.

Have good fun with jax, he is really a good character and i'm going to promise using this build (or something similar) will get you good results, tell me what you think and good luck!