Jax is melee bruiser and a dps. he is hard in the beggining(young game) but easy to play at the end of the game(late game). He is very strong 1x1 but he cant destroy trindamere or nocturne so easy.Jax is a hybrid, that means that he uses as AP as AD. I will tell you what i did to play this champion well.

Runes are slighty different in this champion. Jax needs magic penetration for his skills. I`ll recommend GREATER MARK OF INSIGHT and GREATER GLYPH OF INSIGHT for magic penetration. Buy GREATER SEAL OF EVASION if u want some dodge chance and of course GREATER QUINTESSENCE OF EVASION. Dodge runes are expensive but they will give you about 10% dodge.You can try to putt attack damage or ability power but M.P is the best thing.

Masteries are very complicated. It depends on your jax play.
I will recommend to fill attack zone; you putt 3 points in ARCHMAGE SAVYY and 1 point in CRIPPLE. Now you can go to SORCERY, where you putt about 4 points.that will lead you to ARCHAIC KNOWLEDGE, that gives you 15% magic penetration, i highly recommend.You can now fill ALACRATY that gives you attack speed, sunder that gives you armor penetration and brute force, which gives you 3 attack damage.The next masterie is lethality, where i dont recommend so much in case that you dont have critical rate yet. Now you finish with havoc , the best masterie yet, it gives you 5% increased damage, as magic damage as physical damage.
You now go to defense zone, where you will putt hardiness at max,(gives you 6 armor) and 1 point at resistance, which gives you 2 magic resistance.this is just to get you to the next masterie, this is not so important.
I highly recommend to max evasion, becouse it gives you 2% dodge.VERY IMPORTANT !
Now use your last point in Nimbleness, to let u escape more quicly.

You can start with a DORAN BLADE, doran shield isnt so effective with this champion, becouse of your passive.That gives you health for attack damage and ability power.
there are only one boots, the ninja tabi, this boots are important becouse they give you 12% dodge chance and 25 armor, giving you more chance to stun.
After these cheap items you can start doing the guinsoos rageblade, this is an hybrid item, so it favours your abilitys by boths sides.this particuly item also increases your attack speed, taht will stack with your passive of your ultimate, while attacking.
You probably have these itens in the beginning of MID GAME. So you can start doing your hextech blade. In my opinion this item is the best for jax in the game, it gives you 20% life steal, to make you a little harder to your opponents while attacking; 25% spell vamp to steal some life while doing abilities; 75 ability power to increase your spell power and 65 attack damage to increase your melee damage.
You are now probably entering LATE GAME, so start making your phamtom dancer, to finnaly give you some crital chance, an high attack speed and some movement speed aswell. You can now destroy one champion 1x1.
If you game istaking a while, you can make a INFINITY EDGE to make your critical strikes a much more effective. It also gives you critical chance and 80 attack damage.
Now you finish your item build with TRINITY FORCE to make your melee attack unstopabble after one abilitie. Have luck making these expensive itens.

You can go to your lane with empower, wich increases your base damage of your next mellee attack. At level 2 putt your point in LEAP STRIKE or in COUNTER-STRIKE, to make you jump to your ennemies or to make you stun the whole game. Counter strike also gives you 10% dodge chance at first point, increasing if you putt points.FIrst envolve empower for hight damage 1x1, then leap strike for hight pursuit and finnaly dodge, to give you and hight dodge chance, of course, you must envolve your ultimate at level 6 , 11 and 16.

Summoners spells also can be different, if you want an easy escape or an high chance to kill in a pursuit, you must use flash and ghost.
But if you want an high chance to kill the enemie in a 1x1 match or 2x2 etc. , you must use ignite and exaust, ignite is a spell that i think that it was made to kill champions like trindamere, dr.mundo, fiddlsticks and warwick. Exaust to highly decrease the target damage, both for skill and for melee or ranged attack. Its a matter of play and game tactic, i recommend, for beginners ignite and exaust.

We can say that jax is a push becouse of his jump, that deals severe damage. Jax is most wanted for players that want damage but also to stand and fight, jax was nerfed when phantom dancer dodge chance disappeared, that it was about 20% per phantom dancer, if you would make 5 phantom dancers, u could have 100% dodge chance. Jax is not a Penta Kill champion, he can do it but its rare. Try to have conflict 1on1 or 2on2, never in a minus number on your side. Things can turn bad on you.

Tell your team-mates to push sides or the middle, so jax can go destroy the lanes tower in peace.
You can also quicly kill an enemie champion when he is at low level, a stun might help you.
Jax is always the last one entering the fight, so he can finish the mages and next the otehr fighters. This way is the most successeful thing to get a triple, quadra or penta kill.
Jax have combinations, he can combine with rammus, becouse of his taunt, with evelynn, by her precious stun, with sona and soraka, so he can be healed fast. Remember to have a good team, if you are premade you must have 2 TANKS,2 FIGHTERS and 1 MAGE. To have an balenced team.

Hextech Gunblade unique passive can be used to pursuit and it dosent stack.
Also has Trinity Force unique passive, that makes your next attack,after an ability used, to make 150% of your attack damage as bonus damage.This effect dosent stack with sheen.

Farming is hard with jax, i recommend to spend some time alone in a lane after mid game.
Farming will get easier after level 6, so you get your ultimate and your attack speed will increase. Jungle if needed, but jax isnt good at all at jungling.

Ranked play will get a little hard with jax, becouse he is 1x1 and without AoE damage ( area damage). This build is so effective as in ranked game as in normal game, becouse its all the same thing. Just go ranked with jax if you really are a pro or you dont have anything else to play.

jax is hard at low level, so dont get mad. When you reach level 11 and you have hextech gunblade by half, you will notice some fire power. Have fun and good luck with this melee bruiser.

Cons
:nice pusher
:can easily kill
:hava high dodge chance


Pros
:dosent makes AoE damage
:easy to kill
:hard to farm