Jax by Mosh Josh, on Flickr

All my guides are for my own reference, but if someone finds them helpful that's fantastic and I'd be glad for any tips or ideas.

Jax, truly the Champion of all other Champions. Really looking to get a penta-kill? He's your best bet. No other DPS'er thus far truly threatens him.

People complain about tanky dps champions, Jax is the reason why.

Cons
-Slow early game
-Not a good farmer (at first)
-Very little range

Pros
-Almost cant be beat 1v1 late game
-Excellent pusher
-With this build, massive lifesteal
-Fairly good survivability
-Fits most team compositions
-Wins team fights

Why Exhaust and Ghost?

These two summoner spells are both a delight for any auto-attack champions. They can both be used as survival tools to escape fights gone bad or to pursue your next kill.

[u]Viable Alternatives[/u]
Flash - Interchangable with Ghost, but I always choose Ghost on mellee dps champions because it allows you more time to chase and hit an enemy instead of just flashing and hoping to kill them before they escape.
Ignite - In situations were you're up against champions like Irelia, Warwick, Vladimir, etc. to reduce their healing. but since you cant expect to know your enemies I do not choose this spell. Otherwise just to get that last bit of damage in on an enemy.
Cleanse - Crowd control is what really stops Jax from eating teams and to have Cleanse as an ace in the hole is pretty helpful
Teleport - Pretty self explanatory, mobility, turret defense, remaining in lane, and even potentially escaping death.

Useless Alternatives
Clairvoyance - I can think of one use for this on Jax and thats if you plan on running through the jungle a lot and may need a quick escape, so you Clairvoyance a neutral minion and leap to it. If you plan on doing this and you are level 12 you need a good slapping. Use Flash
Smite - Jungling is not common for a Jax, I can't see this being helpful
Rally - Does anyone use this?
Revive - Also ^

I believe its very common practice to take Leap Strike first because it can be a tool to pursue enemies or jump away to safety. Afterwards we take Empower for that massive bonus from combining Q and W on an enemy. Then a point in Counter Strikefor the additional 10% Dodge which is crucial to his victory.

Skill focus
Relentless Assault
Empower
Leap Strike
Counter Strike

The reasoning behind focusing on raising Empower instead of Leap Strike is the vast difference in mana cost and it gives more damage per level. The only advantage in pushing your Leap Strike levels is to have your leap on a lower CD.

Jax is an excellent bruiser with high damage output, therefore I put my 21 into the Attack Tree because an extra 5% damage when you hit 2.5 times a second for massive amounts does add up.

The 9 in the defense tree is for one purpose. To unlock Nimbleness . This is ESSENTIAL on a good Jax, with a Dodge chance at 40%, in fights or near enemy minions its like having a constant 10% speed boost.

Jax is well suited to initiate in teamfights as late game he has fairly high health. Most importantly though, that way you will nearly always be able to activate your Counter Strike right away. This is what allows him to take on 4 enemies at once, his ability to stun all enemies in his vicinity.

Time for my favorite part

Now all champions are item dependent and if built wrong become worthless in a game. Jax is no exception and in fact it is even more important for him to be built correctly.

Relentless Assault is Jax's passive and it is what make him even more incredible as a hybrid. Stacking AD and AP give him health? Its like receiving a free Warmog's Armor late game while not losing out on building up your damage output.

The Items

Ninja Tabi - Its common knowledge that every Jax needs these boots and no others, 12% Dodge chance is a crucial boost for him.
Phantom Dancer - Attack Speed is absolutely a must on Jax because it allows you to use your passive from Relentless Assault more often. Not to mention a movement speed boost is always nice.
Guinsoo's Rageblade - I can not emphasize the importance of this item enough. It provides both AP and AD, both of which add to his overall health, but more importantly this item couples with his Ultimate so well that around lvl 13 you can already have damn near 2.5 attack speed. It allows you to literally eat turrets and enemy champions if they stick around to fight you long enough.
Hextech Gunblade - Now you might be asking right now... why 2? The answer is 2 Gublades allow Jax to live through nearly anything as long as he is hitting. One gives him 330 additional HP, thats just one! Even better than that is 40% Lifesteal & 50% Spell Vamp from having two.
Frozen Mallet - I feel like this item with give me a lot of contraversy as most people are pushing the Rylai's Scepter on Jax, but I do not. He is a melee DPS end of story his ability to repeatedly pummel should be the way he keeps his enemies slowed. Granted I realize that Relentless Assaults hits proc Rylai's I'd rather do it every hit than every third.

Not sure what to say, just early game take it slow and don't get greedy, last hit minions and once you have a Phantom Dancer you should be set to start taking on enemies 1 v 1 and eventually 1 v 5 if they're not organized.

Happy Killing and see you on the Rift.