Hi, Im not really a high elo player, but this is probably gonna be a decent build, I use it all the time and today I wanna explain why I build Jax this way :P

Flash I build flash because I use the Stun flash combo when I already iniated with my Q, they either Flash away/TP or they're to fast for me, I've done it plenty of times and succeded

http://is.gd/ee1jN2 Ignite Well, its usally common with jax that you pick up Ignite . Its basically your finishing move at early levels (1-6),its not hard to finish your opponent with flash, most of them use the healing potion too late, but its always a bet.

Alternative:

Ghost Excellent for chasing, in this case you will be maxing Q instead of W in that case your using ghost,

Leap strike Its good abillity that does a decent amount of damage at lvl 1 but its not gonna be useful till lvl 3 (IMO), you've got flash so thats gonna ensure some kills then and then.
Why? well you got your stun and you can basically eat them at lvl 2 with your Empower and ignite, it doesn't simply do more damage, it opens more possibillities for initiating with leap strike at lvl 2 sure, but this is my choice :)

Empower This skill, man I love this skill.. You can either use your Empower combined with your leapstrike and get some burst off or ensuring kills :)
But I use Empower for the huge damage output, it resets your attack timer when you use it, so 1 basic attack and one empower will deal your normal damage aswell as your empower damage,

Counterstrike Basically stuns your opponents/opponent after 2 seconds of dodging, Press E, then wait the duration out and your gonna deal damage aswell as stun them for about 1 second (The stun duration will not increase if you level it up but the cooldown will be lower)
I choose this skill at the beginning beucase you can dodge every basic attack, he also gains 25% AoE damage reduction( AP or AD doesn't matter) that makes him a little tankier when counter strike is activated.

Grandmasters might
Passive: holy **** this deals alot of damage, every third attack jax deals a additional 100 magical damage (plus some of your ap)
Active, you'll gain Armor and magic resistance while the abillity is active, it stacks with your AD and AP, so if you wanna go AP you'll gain moremagic resist, if you wanna get AD you'll gain more armor.
This will also apply if you want to go more tanky Armor will give you AD and Magic Resist will give you AP, its really a win-win situation.

I WILL POST A VIDEO HOW I BUILD JAX ASWELL.

This is how I build Jax with masteries, I don't know if its the best but its how I build him
There's not really much to talk about in this except you'll get some armor of this and some regen aswell when you get low.

Why scaling magic resist? : Well if your against a AP, your gonna play more defensivly, but you'll get tankier by the levels :)

Why armor?: well you know, you wanna be as tanky as you can with Jax at the beginning
You'll have like 36-39 armor, Im not sure cuz Im not in the league of legends atm :S

Why Attack speed Quins? : Well you need more attack speed than you think early game so that's why

Why Damage Quins: Why not? Its like 68 damage you'll have at lvl 1 I think :S

Elixir of fortitude: This will give you 15 damage and 125 health for 3 minutes when you consume it,you can bait ignites with this, even towerdives if they're stupid xD

3HP-pots, its a choose things, if you know that they aren't ganking so much, your fine with 3 HP pots then but pots are probably the way to go

1HP-pots and 1 ward(I think :S )
This isn't very solid either but it works if you watch the map alot, or if teammates know that they are coming top to gank or if your getting ganked, its your choice!

(optional rush) Bilgewaters cutlass: you're gonna build this becuase it gives you a good amount of damage to compete with and it also has a slow so its really benefitial to buy this item, you can secure kills with it and ignite

Boots of speed
Mecury threads, If you know that they are AP and they've got stuns as ****, you'll probably wanna go Merc threads :)
Ninja tabi: If they're more Attack damage based team, with alot of poking potential get theese and some HP pots :p

Blade of the ruined king: Its just the way to go IMO, I don't know why they go hextech gunblade, it'll just give you some AD and some AP with Spellvamp and lifesteal, your not gonna go more AP based this build but it work(getting of topic ...)
ANYWAYS it gives you a huge damage buff against more bulky opponents with alot of HP, Each hit will drain 2.5 % of their current health in pysichal damage, your gonna be a hard opponent to beat if they wanna go 1v1 against you.
The active abillity is awesome, it slows your targeted enemy, heals you for the 15% plus some of your AD, which is alot of health and heals you with it, it'll also steal 30% of the enemy's movement speed for 4 seconds giving you more chasing potential :)

Warmogs armor: You'll be really tanky with this item, it'll remove some of Jax's squishyness,
it heals you 1 percent of your maximum health but when you get hit by a enemy opponent it will cancel and you'll have to wait some additional seconds for it to activate again (just stay out of combat if you want to regen some much needed help)

Phage: More slows please! Slows your enemy for 35% of their movement speed aswell as the attackspeed (that with active abillity from the blade of the ruined king = op slow)

Sheen: its like a crit basically, it give you mana and procs when you activate a skill every 2nd second.

Trinity force, just one word! TONS OF DAMAGE!, like seriously, its gonna give you much needed health from the Phage you bought, your not gonna purchase the zeal because the recepie costs like 3 gold or something :) Sheen gets upgraded from 100 % of your damage to 150% of your damage, so its basically like a infinity edge :) plus it procs every 2nd second

If there is a late game you'll probably wanna get tankier so here's some items to choose :)

Guardian angel: 50 armor and 30 magic resist plus the 30 percent of your health and your mana when you get revived, gives you more survivalbillity and getting more tanky

Sunfire cape: if your in need of more HP you want to get armor and health then get randuins omen seriously, because sunfire cape isn't really good on Jax (IMO) But I've seen people getting this and succeded very well

Randiuns omen: 70 armor and 500 health what could go wrong?
It also has a passive and a active abillity
Passive: everytime a enemy hits you, you're gonna have 15% chance to slow them down (attackspeed and movementspeed) which is really good against characters that are squishy, because you can catch up to them
Active: You'll slow every enemy within a 500 unit radius and slow them for 35 % of their movement speed (Idk if attackspeed aswell)

Runic bulwark : If someone hasn't picked this up yet they are pretty stupid. In any case you can get it for your team, it gives you health amror and magic restis, which is still much needed :)
It also gives out a aura that will increase you allies health regeniration by 10 per 5
It will also give this to minions but half the amount

Maw of mawmortious: I have never used this item but it could work (not gonna recommend this, ITS OPTIONAL!) but its good if your against heavy magical damage enemies, it gives you a 400 health magical shield that protects you from magical abillites such as, veigar dark matter ( I think o:)

More damage? Kay

Get Atmas impailer : Awsome, gives you 1.5 % of your maximum health as damage, so if you have 3k hp its gonna give you 45 damage, so its basically like a BF sword
Zephyr (Imo its not even needed, you already have alot of attack speed, no need to get more, aswell as damage if you get Atmas, but the movement speed bonus is a cool add!)

I think this is a good guide myself ( cuz I made it LOL ), its gonna get you to silver or even higher, I think :)

Pros and cons: I got some pros and cons

Pros:
+ Your gonna get tanky as a fat chogath
+ Your gonna hit hard
+ You can save teammates from sticky situations
+ If you die first in your team, no worries, The GA will keep you alive to do some damage afterwards.

Cons:
-Really weak early game
-Needs farm to be really sucessful

If you find this guide good or bad, leave a comment and tell me your opinions, its my first guide and I hope you enjoyed it! :)
If you wanna find me on lol king here
http://www.lolking.net/summoner/euw/20893802
Not a high elo player as I said, but it is for funs :)

But give me some proffesional oppionions :) bai