Not Updated For Current Season
First attempt at making a guide...
Jax is my main and is one of the first characters I got.
First Back: A normal first back is at 1000g to get Skirmisher's Sabre and boots, but if you go on a killing spree then you want to rush Devourer without boots.
Second Back: You want to have 1400g on second back to go get Devourer. Get it to Sated ASAP.
Third Back: Depending on how bad life steal is needed, I usually always rush Rageblade. I sometimes will get Cutlass, then Rageblade. Anyways, you want to have enough gold to get Blasting Wand and Pickaxe. If you are doing really well, then try to get enough gold for Rageblade.
Fourth Back: You want Cutlass right after Rageblade. Also, try to get Hextech Revolver (if not get 1 or 2 Amplifying Tome(s)).
Fifth Back: If you haven't bought Hextech Gunblade, get that. If you have Hextech, then buy items to build up to Sterak's Gage (starting with Jaurim's Fist).
Last Item (Trinity Force): I now get this item last. This is due to the cost of the Tri Force. You want to buy one of the 3 items: Phage, Zeal, or Sheen. Which one to buy just depends on who you are playing against.
Hextech Gunblade is my favorite Item on Jax. Ever since RIOT made Rageblade better I started to get that first. Hextech + Rageblade + Tri force + Sated Devourer = OP burst on 3rd basic attack because of passive. I really like Sterak's Gage on Jax. It goes along really well with certain supports or other champs. I usually get tri force last item now because of how expensive it still is.
Try to always carry a more ward other than your trinket. It's a life saver.
Creeping / Jungling
Start at golems on red side. Gank at lvl 3, if possible. go get blue or all of blue side and then try to gank again. If you can't gank again, make sure you have at least 1000g to get jg item and boots on first back.