This is how I play jax and I almost always come out with more kills than deaths. This guide is good for beginners and can be modified for more experienced players based on play style.

PROS:
-Able to tower dive and escape easily
-Great DPS with Guinsoo's Rageblade and Relentless Assault stacking attack speed
-Great carry
-Can take down towers quickly
-Great at running down enemies

CONS:
-Weak early game
-Targeted when leaping into battles
-His stun only works when he dodges

I like to have my Attack Speed to help with early game kills and once I have guinsoo it helps stack the attack speed making the damage output just that much higher. It becomes equally useful once you have Vampiric Specter to help heal faster, and even more so when you have Hextech Gunblade. The dodge runes are a must have on Jax, his stun relies on the dodge, just has to be there. Now when it comes to the Glyphs, you can change it around accordingly to what you feel is more useful, I like Mpen simply because I like my initial damage when I lunge into battle or I go for a tower dive it gives you that little extra damage output you need sometimes.

Masteries for me depend on who I'm playing with. If I see my team doesn't have a very good tank or just a bunch of melee's (Garen, Xin, Jarvan, etc.) I'll usually keep it how it is above. Its also good to have if you're planning on diving alot or going first into team fights.
Otherwise I like to have my masteries set up as 19/11/0 with the armour penetration in the Offence section, while keeping Nimbleness in the Defence section. It all depends on what you like, more damage or more armour.

When it comes to my first items its always the same.
, 5 hp pots.
always 2nd, by the time I go back for this I already have Counter Strike, so the 12% dodge from the boots is needed to activate the spell/stun.
Next I get , the extra damage you get on your initial jump with empower is great. Also you won't be able to jump alot at the beginning so you need the increase in your basic attack to finish enemies.
After , I usually get but if you find you're getting beat and recalling too much, I would get specter helps with healing allows you to stay out longer.
Once you have , killing should come pretty easily now. The way it stacks with Relentless Assault just makes your attack speed insane.
Now you should be getting Specter to help with heals and keep you out in lanes longer.
From that point on you can build which ever way you like, I like the because it makes your empower stronger while giving you spell vamp. But is good if your just holding a lane while giving you lifesteal to heal.
After , most people who build Jax will disagree with me, but I like the .
just help take out those characters who are too fast (Master Yi, Teemo, MF, GP, etc.) by slowing them and gives you more survivability to tower dive and take towers while giving you a little extra damage.
After that I would probably go for Banshee Veil because people start targeting you because of your insane damage.

**NOTE** If you're getting targeted too much you might want to get Banshee Veil before Frozen Mallet

My skill sequence can vary, more towards mid game. If I find I need to jump more ill go for Leap Strike to reduce the cooldown. But other than that, I always do it in this order, the Empower cooldown is quicker than Leap Strike allowing you to put out more damage if you level that more than Leaping Strike. Counter Strike is useless to level up more than once early on, only need 1 stun usually to finish someone.

Exhaust and Ghost are what I always take, but depending on how much you want to tower dive you can go with Flash rather than Ghost to get out quick. But late game Ghost becomes really effective at running down enemies by stacking with Nimbleness once you dodge, then its just a matter of time before you catch up to your enemy and leap at them. Exhaust to me is a must have with the reduced armour and magic resist from the masteries it makes your attacks just that much more deadly.
Everything else to me isn't worth it on Jax.
Some builds will try Cleanse and Ignite, but the Ignite is only really good early game which I don't recommend trying to fight right away, so its a no for me.
As for cleanse, I would only use it if you're really good at leaping away from combat, for most people it isn't easy, so its a no for most.
As for everything else...just no never.

When you get mid game and people start ganking, NEVER be the first one in battle. I will garantee you that you will die if you do go first.
Always let your tanks go first or let someone get a stun/slow on an enemy first, then empower, leap strike, counter strike and the enemy should be dead, or almost dead.
NEVER aim for tanks first, once you've killed a mage or dps char, you will have stacked enough attack speed from guinsoo's and relentless assault to then try to attack tanks, I still wouldn't suggest it until the squishies/dps characters are all dead.

This is what you can expect as end game results with my build...
Please vote/comment this is my first build, probably missing little details here and there I will be adding them as I remember then or as people comment :)