I'm not a pro, but I'm certainly a Jax fanatic...
I have only played about 70+ games but I played most of it using Jax.
So here it goes...
Leap Strike: Jax leaps toward an enemy unit and smacks it in the face with his lamppost. Jax does a leaping attack to an enemy dealing an additional 50/75/100/125/150 (+1) magic damage.
This is, for me, Jax's core skill. This skill would let you, not only harrass your enemy, but also gives you a lot of potential for an ambush. It's also very good for chasing down fleeing enemies. Should max this as soon as possible.
Empower: Jax's charges up for a cleaving attack, dealing additional damage to a wider range the longer Jax waits to strike. Jax empowers his weapon, charging up the area of effect and 25/35/45/55/65 bonus magic damage each second for up to 3 seconds. His next attack will splash, dealing 60% damage to nearby enemies.
As weird as it may sound, I'm not a fan of this skill. It's quite a good skill, with a high bonus and only needing a small amount of mana. But, when in lanes, it makes your next move predictable coz the enemy would see your lamp glow. Most ppl will argue, "Use the brushes", but if you have a decent opponent, they would also use the brushes. plus, this skill can be hardly used on a sudden situation. The skill is good, but I rather get Counter strike first.
Counter Strike: Jax's prowless allows him to dodge enemy attacks frequently. After avoiding an attack Jax can quickly stun surrounding enemies. Passive: Increases Jax's dodge chance by 10/12/14/16/18%.
Active: Jax stuns surrounding enemies for 1 second and deals 80/100/120/140/160 (+0.8) magic damage to them. Usable only within 7 seconds of dodging an attack.
Instead of Empower+Leap Combo, I use Leap+Counter Combo. How? The technique is to let the minions hit you to activate counter strike then leap to the enemy. Damage-wise, it would deal more damage than Emp+Leap since if you stun an enemy, you would have the time to do another autoattack before he wakes up. It's also AoE which makes it really great.
Relentless Assault: Jax's greatest talent lies in his ability to grow stronger with every attack. Each successive attack increases his attack speed temporarily, and every 3rd blow will devastate his target. Jax's basic attacks increase his attack speed by 6/10/14% for a short duration, and hitting the same unit 2 consecutive times enhances Jax's next attack for 140/170/210 (+1) bonus magic damage.
Attack speed stacks up to 10 times.
This skill is what makes it possible for Jax to 1v1 other champions. And this is what makes Leap+Counter greater. When you stun your enemy, you would have some time to hit a couple of times good enough to activate you ulti.
Some ppl might be wondering how would you activate Counter Strike when ambushing. It's quite simple, when the enemy is beside his minions, if you target him the minions would start attacking you. Then, dodge, bammm Stun.
I start my item with boots and a couple of hp pots. Being able to move faster would help you a lot on last hitting minions, especially when the enemy is trying to harrass you. Then get Ninja Tabi as soon as possible.
Blasting Wand. This would be use to purchase Rageblade later. And why BW over pickaxe? Because on early game, you would be executing skills more often than your autoattacks. plus, it gives a fair bonus to 3 of your skills. Buy a Rageblade as soon as possible, it gives a fair amount of wellrounded bonuses, AP, Atkdmg and Atkspd.
Phage. This would give you an ease ganking heros on your own. This will renders them from running away from you, plus the extra 225 health that would always come in handy.
Phantom Dancer. This will make you almost invincible to autoattacks. A total of 42 dodge, together with Ninja Tabi and Counter Strike. Also, the added movespeed is very useful more than you imagine it would be.
Believe or not, this next item is a plausible item for Jax. Frozen Heart. On team fights, it's not enough to just relentlessly bash an enemy. Because, 1. They do the same to you plus disablers, and 2. They move around all the time. Frozen Heart will let you be able to jump from enemy to enemy. And mostly, it would able you to spam your stun more often.
Lastly, get bloodthirster, this would serve as your damage booster and lifesteal. But if you found yourself getting focus fired all the time on a team fight. you might wanna consider buying Banshee's Veil or Guardian Angel instead.
I'm still a low level so I have a limited mastery points. Investing 9 points in Defense would give you Nimbleness. The rest could be use on utility for regen.
I don't have a large experience on runes so I won't comment on it yet.
Helps a lot on early to mid game ganking. Would give a big advantage on Level 1 Ganking.
Great for Team fights. Excellent for saving an wounded ally. And also for saving yourself.