Old School. Double Bloodthirster. That is all.

Jax in the jungle is not really that great. Sure he is strong, very useful, and can make great plays, but he is a kill stealer. If you don't know, you don't want to take kills from your lanes. If you can you should start with a ward, boots, and pot then plant the ward on the enemy red if the enemy jungler starts blue. kill tehm when they take it, since they will most likely be low on health from other jungle camps, but be careful since they might be ganking and you don't want to waste time with jungle Jax since all you want is your early items, then it would be easy to gank and get you and your team fed.

Is bad

I made the New School Jax build just to show you the item build. I don't use it (sometimes I do, but almost never), but i hate seeing people build Jax Ad. , and have Hextech Gunblade. If you haven't noticed, the active on Blade of the Ruined King is very similar to the active on Hextech Gunblade. Hence, why it is so efficient to build Jax so.

There is not one build or way to play Jax. Mix what you see, you might be better off that way. Oh and don't forget your elixirs, they are op.

Also, the reason Jax doesn't need a real weapon to kill someone, is because he is from splinter cell and is an assassin.