First build is my typical HYBYRID build.
Second build is my typical conservative build.

Go conservative if you know your being denied of farming
if not hybrid it is.

My ingame name is n169erc0ck. Please if your adding me tell my why you are adding me, or else im just gonna block you.
Ive been leaching off mobafire for like maybe 3 characters or so, now I realize I could contribute the this community with my own jax build.

Pro
Easy carry killer.
Doesnt die easily
Cool drunken man dance
best 1v1 champ in LoL
Con
Public enemy
Needs to have some early game gold or great farming to shine.
Takes alot of gold to start rolling.

Doron ring
Doron blade

or...

cloth+5pots.
Boots +2pots

I dont recommend Doron shield.....unless you soloing..But in that case blade is better.

AP quintessential runes are probably the best because of jax passive of ap converted to HP.
+26 health runes arent that great, but they work pretty good still.

Yellow Armor also can do pretty well. Dodge is preferred.

Blue AP for maximum amount of AP.
Blue magic resist also works too..cant ever have enough mr.

Brown armor/magic pen. If your going hybrid.
Ad use armor pen, AP use magic pen
Brown AD runes are not bad too. .95AD gives u about 25ish hp?

Standard jax is 9-9-12 which is pretty balanced.
You could go 21-9-0 but that would be a sacrifice of EXP, unless you are duoing with a zilian.
In my opinion 9-9-12 or 9-21-0 are all good. I wouldn't recommend 21-9-0, unless you have some really good runes.

Best spells

Flash (Getting in or out. Chasing or running)
Ignite (great for drawing first blood. FOr the extra AP. Or for stopping that mundo ult regen.)
exaust (makes you rock in 1v1, or make the other person get away from you.)
cleanse (CC is the only thing that really kills jax. And this can help you alot.
ghost (Ghost is great for chasing or running away, since you can leap strike you are a really good chaser)

Other spells. Dont think about it seriously..

WHY FON?

Great magic resist AND good regen. I usually choose this over banshee. With build one you should have about 3k+hp and you need the regen.

Why Malady
Gotta love the Magic resist shredding. The damage and AP from the item. Not to mention its pretty cheap.

No ninja tabi? No problem.
If you have full dodge runes. all yellow and quintessential dodge you have enough dodge from that, so you dont need ninja tabi. I recommend tenacity boot OR magic pen.

(Thinking about list)

rylai crystal scepter gives 500hp and 80ap+slow. I find that its a great item I get it if the other team is really mobile..

stacking hextech gunblades
I actually think this is a viable idea, if you are stomping so hard that you have that kind of money...its a great idea, just expensive...

Quicksilver sash
Great item, removes CC and cheap.

menjai soul stealer
jax is a tough character and hard to kill if he isnt CCed. Great snowball item and he really needs the cdr. But make sure you can keep the stacks before getting it.

Situational (kinda recommend)
Void staff=magic pen
cloak and dagger. AS+CS+tenacity.
Madred bloodrazer. Shreads high hp character. I never use it on jax, don't find it really great on him. Unless you want to solo baron or something...

Maybe list
atmas impaler.
Yes jax has alot of hp, but I never get this item because I find it a waste of a slot, and building ap instead is alot better.

I play very conservative, I dont give the other team any reason to come down and gank. I try to stay in lane and just farm. I try to absorb as much exp as possible. but when im done with hextech revolver and lifesteal, I usually go and jungle so my partner can earn more exp.

At this point Im usually jungling, I dont bring madred razer, although I should. Just hextech revolver and vampiric scepter is enough to jungle. I usually only stun and empower attack them because q takes too much mana, (unless you have blue buff). I do occasinally gank, but only if my lane partner is able to solo.

Mid game is also where jax dominates, If you ward some spots, you can take advantage and exaust them and then 1v1 them. I rack most my kills here, but remember PUSH TURRETS! WHENEVER!!

Usually the case is not to intiate team fights. But if its 1v1 all you have to do is get dodge proc up. jump (autohit) stun (autohit) empower (autohit) and finish off with another hit or if not use hextech gunblade unique active to chase them.

In team fights, I aim for squishy. Like caitlyn, ashe, kogmaw, etc. Its usually an assassin job to clean them up. But they are the main damage dealer, kill them first.

But later ingame the other team will get smart and cc you first, kill you and then go on with killing your team.

PURE AD. (Didnt finish with the BF sword..too lazy to go back to base :P) Solo baron no problem. No buff needed too =D.
I will put some up of my win of all jax, because lately I have been playing some free characters..