Smite!! -- Needed for jungling, you could do it without it, but then jungling would take to long and you would miss out on a little gold you get with smite if you get the mastery of course. THIS IS A MUST

Ghost -- allows you to rush right in and make it near impossible for the enemy team to get away from you especially if the are way too aggressive

Exhaust -- It works but it at most can guarantee that you kill one in the lane but the other will get away can also save you and make the other champion weaker so you could stay in a battle that would be suicide under other circumstances

Flash -- Works allows you to close a gap between you and an enemy quickly so you can get the jump off or stun them, but it isn't a lasting effect so it doesn't always help in the chasing role but it can help you escape very well.

Other Spells -- if you like it then go ahead and try it but I have found that other spells DO NOT help much at all when it comes to jungling.

The runes that have been chosen were to get through the jungle as fast as possible and to also make sure that you don't get to low in the jungle in case of ganks.
AD marks to kill the creeps faster
Armor seals to lower the damage
AS Glyphs deal damage more often
LS quints to make sure that you don't get low while you are jungling so you can gank right away if needed and to make sure you can survive in the jungle if they choose to gank you.

The Vamp scepter is to make sure you can maximize survive-ability in the jungle in case of a gank.

Boots of speed as soon as you blue pill so that you can keep up with enemy champs when you gank

Wriggles Lantern for more life steal and so that you can destroy the buffs a lot quicker so you can gank more and do more damage. Also gives free wards, and you could use those wards to chase someone or to go over walls when escaping by jumping to them

Berserkers Greaves -- p the attack speed and make sure you have more speed and attack speed to destroy the champions. Also it is very cheap and with junglers they dont get as much gold as laning champions do unless you are ganking really well and getting a lot of kills.
Mercury's treads -- Also great boots for Jax because it lowers how lost cc can work on you and make it extremely hard for the enemy team to get away from you when you gank


Next you can go for Trinity Force or the Hextech gunblade if you are doing well then go for the trinity force for more damage it really ups the damage by a lot. But if you are not doing well in the ganking department go for the hextech for more survive-alility it increases the passive of your ult, increases the damage of your w and you can you the active to slow down the enemy to make sure they can get away

The rest of the items are completely dependent on the game itself.
Guinsoo's rageblade -- A great choice for jax because it really ups the damage he deals and his attack speed by an amazing amount

Frozen Mallet -- Great for enemies who are really good at escaping and you need to chase them down, yes the trinity force gives a little slow but it doesn't give as much as frozen mallet and not as consistent, and it gives more health to make you a lot harder to kill.

Maw of Malimortius -- Another great item because in game people are not going to like you so they will target you and the maw gives magic resistance as well as a great amount of attack damage so while you are fighting you can use your dodge and your damage will be increased because of the health you have lost, your ult will make your armor go up because you are gaining more damage and if you get to low you get a nice magic shield to help cover against the enemy champions you cannot dodge against.

You could get Warmogs and Atma's Impaler which I have seen a lot by other jax's, it does work for him it makes him harder to kill BUT it also lowers the amount of damage you could potentially do. With Jax's ult that ups the Armor and Magic resist you don't really need a tanky build because the ult itself makes him very tanky and you can also dodge to make it very difficult for the enemy team to kill Jax. Also with the tanky build your ganking suffers because you do not deal as much damage early game

Basic Jungle route first

Blue
Wolves
Wraiths
Double Golems
red
Blue pill
Buy boots
gank!!!

Don't gank before lvl 3 to make sure that no one can get away from the ganks

Destroys minions really quickly but because you are jungling you are not offered as much farm as the lane would offer so your gold income is mainly based on how well you can gank

Jax is very good at initiating in team fights, he can junp in an stun the whole team while avoiding damage at the same time to help your team jump in and destroy the opponents. He is also very good at chasing down enemies and slowing them down for your team to get there, if you don't kill them yourself of course.

Pros

Does a lot of damage really quickly strong both early game and late game
Has a jump to get into peoples faces for ganks
Only character in game that can dodge
Can become tanky by building up damage of any kind
Plays hybrid Extremely well which makes it harder for tanks to build against
Thornmail pretty much ineffective because of how much life steal you have and you deal a lot of magic damage
Very easy to jungle with
Can still attack while dodging

Cons

squishy early game
cant lvl 2 gank because all moves are really needed to gank with him
not very fast with out boots
takes time to build up his attack speed
Magic damage destroys him because he cannot dodge it
Needs to hit moves to ensure the gank
Low range stun

This is just a build that has worked very well for me as I have played Jax. It is not a pro build by any means. I have not tested this in ranked play, but I will. This is my first real guide. It has only been tested in normal play, and it works really well there. I will update this build as time goes on. Any suggestions on how I can make this a better build would be greatly appreciated.

-- Jax can initiate but don't get cocky no matter how fed you are you are not invincible. Before you go in make sure your team is willing to go with you.

-- Very easy to get cocky with Jax just play smart with him he's strong not unbeatable just play smart and its game over

-- Finally don't turret dive you cannot dodge the turret it will destroy you!