Basic Jax build, he can be a physical carry or just a hero killer... a little short of Yi in towering but certainly better than Ashe, etc as a laner. The more physical opponents he's facing the better he gets. He's great early game, levels 2-5 to get quick ganks in and even better mid game. I use all attack speed runes for red, all dodge for yellow, and all cooldown for blue. I use 3x dodge quintessences.
I start off with boots and 3x potions, then upgrade to ninja tabi as quickly as i can. Following that I get a phage and then malady. After you have those two farming becomes very easy. I go straight for upping the phage to a trinity force at this point. My next two items are always phantom dancer and infinity edge. I prefer the infinity edge first but if i happen to go back to town with 1200 gold i always pick up a zeal.
Always make sure that you're getting exp, time spent away from fights really puts a dent in any lead you make with kills. After you get the malady you can use the jungle to heal by lifestealing creeps. Keeping jax equal to/ahead of other heroes in gold and levels is especially important because with just a slight lead you can 1v2 melee heros mid game. Try to make sure to get a kill on anyone laning a bit too far every time exhaust is off cooldown. Jax is almost unbeatable by a physical hero in equal gear once you break 30%+ dodge so don't be scared of fighting with 1/2 life vs a full life hero if they don't have spells to burst you.
As far as harassing goes, as soon as you have mallady you should always be trying to tank creeps. As soon as you land a dodge, leap strike and stun the enemy hero. Making them go back to town often is almost as good as farming them as it denies them exp and gold.