Start with a dorans shield for armor and health, the runes should provide enough damage for the start. Grab a pair of boots and then go for hextech revolver. Finish off the boots with either Ninja Tabi or Mercury's Treds depending on whether the enemy is mostly Ability Power or Attack Damage. Now buy the Hextech Gunblade as soon as your boots are finished.
Now buy Guinsoons Rageblade, this will give you lots of attack speed, damage and ability power (this works well with your ult and passive and is a MUST HAVE for jax).

Now you should have a few kills by now but if you don't that doesn't matter, your going to have to rush Trinity Force to get an advantage over your enemies. Now get a Rylai's Crystal Scepter for the ability power and health boost, the slow from your ability's will make sure the enemy doesn't escape. Alright now you will have to buy Rabbadons Deathcap as this will give you a large Ability power boost. Okay, if you are up to this part of the build you are usually approximatly 40 minutes into the game maybe 30 depending on how many kills you have, but now you should buy Warmongs Armor, this will allow you to be tanky while dishing out lots of damage.

Pros
- Do lots of damage
- Don't need to worry about thornmail
- Lots of Health
- Can 1v2 (Depending on what champions)

Cons
- Not much armor or magic resist

A good combo to do is R, E, W, Q, if your ult isn't ready then just E, W, Q, and then autoattack and when you are in a 1v1 against an attack damage champion use your E as much as possible but don't activate it straight away as this will allow you to dodge more attacks before you stun.

If a team mate is in trouble and is being chased, press E (stun) and then W and then Q, but if you are low on mana just E then Q, but if you are REALLY low on mana just run up to them and press E.