Jax is a champion which is very very strong. But he has to be played in a right way.
I don't want to say that this is the only way working but in my eyes with this guide
you are strong enough to take off squishy carry's supporter or Tanky guy's like Garen
and tanky enough to stand those Ap carry's which try's to make your life harder.

Pros:Amazing strong.
You are a counter against Round about everything.
You will make insane dmg while chasing without any high CD
You can stand everything as long as you dont Initiate and dont focus the wrong guys.
Cons
You have to know when you can attack , when you should go back.
You will get flamed so hard you never got flamed for playing a noob char
or you cnat play really good so you have to play jax! (at least they are dead not you
so you shouldn care (; )

The good thing about Jax's Runes is: he is very flexible.
The only thing i would take for sure are the MPen Red. You may ask:" Why Magic penetration ? he scales SO good with ad?!" Yes he scales for sure good with ad but all skills make magic dmg so Armor Penetration isn't so helpfull caus it dont make Your skills hit harder.
For Yellow i take Mana Reg. Dodge is helpfull too and many prefer that but i take the solo lane very very often with jax so i would prefer the the Mana reg
For Blue i take Ap these Runes combined with my Quints gives you right early with 1 jump that O_O OH MY GOD DMG SO NOOB CHAR effect for what are u just playing for. You could also take MR blue if u dont feel well without so much MR but for that u have your Ulti ;)

For Masteries i take 9/0/21 improved magic Penetration Improved Buff Duration pretty general.
I also tried other ways 21/0/9 or 21/9/0 but in my Eyes nothing works as good as 9/0/21

So now the Items I start with Cloth and 5 H Pots. Because i solo often with Jax so i need these Health. If You don't Solo you can also take cloth 3 Hpots and 1 MPot so you can jump at those squishys More often =D.
Build these Armor into Your dodge Shoes and you dodge like Heck.Then you go Hextech guisoon atmas. 1 Typ start always with the Ad items means (Bilgewater-> Hextechgun -> Hextech Gunblade)
(Pickaxe->Blasting Wand-> Guisoons) You should do this because ad scale's harder with all of your skills then ap.
Atmas is 1 of the best items! he fits 2 rolls for Jax.
1.It gives You with More Hp more Dmg and More Dmg means more Jump kills =D
2. It gives You armor! Now you think what are You thinking You dodge like hell!
You have 18% dodge. So You dodge round about 1/5 attacks.There comes atmas the Armor of Atmas
Helps You stand those Tryndamere. Now its on your own:
1. Enemy team has Trynda or some other fed Carry Then: --> Sunfire
2. They have more ap or no fed carry so you dont get much dmg --> Rylai's
Both items give You dmg but Rylai's give you more dmg and that slow which helps you even more
Chasing.

ALWAYS! start with the jump primary for Escaping it can always happen that some greedy gankers are in your bush just wating for you to arrive with your skilled W or E.
now go W then E or E then W its on your own. Now you skill Q and W rotational you make mor dmg then just skillng jump or Empower and also the CD of both will be same so you can jump and empower everytime not just jump or empower.

For Summoner Spells i take Flash and exhaust
Flash is necessary you alway need this flash for chasing/for escaping or for helping those Attacked Supporter in your Team.
Exhaust helps you for slowing enemys like gp or ghost user. If you get attacked or right in a gank use your exhaust always at the most dangerous enemy. You are jax so carry's like Ashe or Mf are not the most dangerous enemy's. AP like Karthus Kassadin or Ezreal are those! They can kill you so you exhaust them if they make to much dmg if they dont make much dmg focus that guy with your exhaust which attacks your supporter.

You are an Assasin. Jump around in the battle get in the battle get off the battle you can do what you want and still make so much dmg! Dont focus the support. Just zone him away than attack the squishys. The good thing here is the supporter can get you out of the fight this can end in losing the fight. so just jump at the support then attack the ap carry/ad carry.

Feel free to counterjungle aggresivly! You kill creeps fast and dont lose much hp/ nothing hp with lifesteal. Most of the jungler are attacker or dont have much dmg skills so you can get away with ease if they find you.

Jax is an Insane character. He is very strong. and has this WOW O_O effect where he takes with 200 hp a full life ashe away. you can make with your jump 1k and more dmg fully build and you tank much. So use your potential =D