Playing AP Jax build, your default rush is Hextech Rocketbelt.
Adapt to an enemy team. Are they full of tanks, and they rush tons of Magic Resist? You rush Void Staff as #2 or #3 item to counter MR stacking. Their team has champs with lots of healing? Rush Oblivion Orb as #2 item (upgrade to Morellonomicon as 4th or 5th item).
Somebody is going after you each fight? Always getting ulted by enemy assassins? Rush Zhonya's Hourglass.
Champion
As Jax playing AP, your skill order depends on a situation. In general, max Empower (W) for damage. Against ranged or for reach, max Leap Strike (Q). Against heavy melee or lots of auto attackers, max Counter Strike (E).
You can use Hextech Rocketbelt either or engage (E + Hextech Rocketbelt + Q to go in), or to run away after your Q + E combo.
Tips and Tricks
Riot
Jax can Leap Strike to friendly units, including wards. You can use them to plan your escape.
Jax benefits greatly from items that have both Ability Power and Attack Damage such as Guinsoo's Rageblade and Hextech Gunblade.