Note: This is my first build on Mobafire. If you down vote it, please give some suggestions as to what I can fix to make it better. Constructive criticism only, please.
The reason I picked Exhaust and Ignite is because Jax does his best as a 1v1 fighter. As soon as you get into a fight, you can blast your opponent with both of these and it will greatly accent your abilities. This also works well against champions with lots of life steal (Warwick, Fiddlesticks, etc.) and helps you kill them faster so you can get out before their team gets there. Some people like to use Flash, but your Leap Strike will serve the same purpose in most situations.
Leap Strike -
This should be your first skill because it will serve two very important early game purposes by letting you escape faster and catch a fleeing champion faster. Later, you can add more just to increase the base damage it does. By end game, when you hit Empower and then Leap Strike onto someone, it will do considerable bonus damage.
Empower - This is your main skill. It has a very low cooldown time and adds significant bonus damage to your attacks. It uses very little mana and should be spammed for getting late hits on minions. Also, when in a team fight or attacking a champion, you should be consistently activating this ability until the fight is over.
Counter Strike - This skill is very useful for its dodge chance and its stun, but you get the majority of the benefit from the first point you put into it, so once we get it at level 4, we will wait until the end to add any more points to it. A good strategy for this skill is to wait until you dodge a minion hit, then quickly Empower, Leap Strike an enemy champion, and follow it up with your Counter Strike. This will do a lot of damage and stun him, at which point, depending on the circumstances, you can finish him off or run away.
Relentless Assault - This is a very convenient ult for Jax. It gives bonus magic damage every 3rd hit (thus the point in the Magic Penetration Mastery) and increases your attack speed temporarily with every consecutive attack. Obviously, you will want to put a point in this every chance you get. The active for Jax's ult is also convenient for attacking AP characters. It will increase your Magic Resist based on your dodge chance (which is a lot) for 8 seconds. I like to use this before I enter a team fight or go in for a gank because it lasts a long time and by the time you realize you need it, it is usually too late.
Your masteries are pretty self explanatory. They are designed to augment your runes and Jax's abilities.
Your main concern with early game is not to gank and get kills as many Jax builds suggest, but to farm as much gold as possible. You should be primarily focuses on last hits with minions. Remember to use Empower to get those last hits on minions with just a little too much HP. Try to stay in lane as long as possible your first time out. Be cautious and try to stay back from enemy Champs. Your 3 Health Pots should do a good job of keeping your health up. Don't return to buy/heal until you absolutely must (which, if you play right, will be a very long time down the road). I tried to set up the best early game buying sequence under the items page.
By this point, you are a viable candidate for team kills. You should have been farming a lot and have your Ninja Tabi, Guinsoo, and Vampiric Sceptor. Try not to get in the middle of team fights where you can be a target. Stay moving on the outside of the fight and wait until your teammates get a few enemy champs low. When the circumstances are right, jump in and try to kill as many low HP champs as you can. Be careful not to stay in too long. Once you get a few hits on someone or get a kill, get back to the edge and wait for another opportunity, or, if you are chasing the team, follow along and try to Empower/Leap Strike them and pick them off. When fights aren't going on, keep laning and farming. Your primary concern still is getting as much gold/experience as possible and with the three items mentioned previously, you can push a lane like no other. If you have down time and the lanes are cleared, you are in perfect condition to jungle. The blue buff golem is always a main target because it lowers you already extremely short cooldowns. Red buff is also great for chasing down fleeing enemies.
By now, you have gotten your Bloodthirster (with lots of stacks because you are farming like crazy) and your Madred's Bloodrazors. You are basically keeping the same mid game strategy of riding the edge of fights and picking people off, but you are doing it faster and easier. Your Bloodrazors also let you take down tanks, since your AS by this point is ridiculous once your ult and Guinsoo get to stacking. By the time you get your Frost Mallet, you should be able to chase down fleeing enemies with no problem. If you make it to Phantom Dancer, you will be able to do everything you have been doing, but better.
Basic Things to Remember:
-If you have your Bloodthirster, are fighting a champ and your health is low, remember, you will be attacking faster and faster with every hit. You will be more likely to survive if you continue attacking (in most cases) then if you tried to run.
-Jax can take down a turret pretty fast by mid game. If your team is engaged and you are far from the fight, try pushing turrets to put pressure on the other team. If they don't start to come after you, then free turret.
-The most important thing you can do as Jax is to farm. This is a very expensive build so you will have to farm a LOT to get your late game items. Always be pushing lanes and always be jungling.
-ALWAYS be spamming Empower and Counter Strike in a fight. Never worry about running out of mana when fighting.
-TIP: If you are having a 1v1 fight against Tryndamere, which seems to happen a lot (I guess they figure they can take you), be sure to save your stun until he is at about 5-10% HP so he will be stunned before he can use Endless Rage and you can get an easy kill without having to chase him for 6 seconds while he is invulnerable.