Hello, my name is Azal, I'm a Jax main. I'm new to this sort of thing, so please bear that in mind when I try to teach you my ways of playing Jax jungle.
Personally, I find Jax to be very skill based, and having a very high skill cap, makes him very fun to play and learn with. Along with his gank oriented kit makes him a very heavy ganker, applying pressure to all of your lanes with every gank, and snowballing your own laners.

Pros:
-Amazing ganks
-Snowballs your entire team
-Fits into a lot of team comps
-Outputs a lot of damage early
Cons:
-Early reliant to snowball
-Relies on team to follow up on ganks
-Sort of falls of mid game and comes back late game

My personal pathing is starting red buff, although the leash from bot side helps, you don't really require a beefy leash if you're placed top side of map, simply starting with Counter Strike nulls the damage from red. And the red buff really helps with an early level 3 gank top/mid. A good way to do this is by starting red, killing krugs, then taking scuttle, this will make you level 3, and you can quickly Ghost into whichever lane you wish to gank, and the e,q,w Thunderlord's Decree combo along with your laner's follow up will be an easy kill early on for a quick snowball.