I can't tell you who I am, but I can tell you I have played Jax for literally YEARS. A long time ago me and some fellow Jax mains broke him down to a science. We discovered the best way to play him as either a split pusher, or a team player.
I'm giving whoever is reading this my skills as I leave League....
I was Gold when I left.
The Jax main who taught me didn't play ranked
But.... he was asked to play on a pro team many times. "Jax mains don't need a team friend <3"

Last thing before you dive in... You picked the best champion in the league friend <3 My summoner name is DeathKnight444 if you have questions... feel free to add me

A "team player" Jax role can be frustrating. You are putting trust into your team (usually 4 people you'll never meet). Regardless your targets are the back line. With randuins, BOTRK and the Phage passive in Tri-Force the adc cant escape you. If they do just Q to them and follow up with a E stun. Kill them and turn back to the rest of the team. Either help them, or just leave. No one knows what Jax is, but if hes not a human or a yordle, or even some weird monster, why help them.

If your choosing to split push as Jax I like you already. Being selfish in this toxic game can get you very far. Focus primarily on farming the ENTIRE game. During laning phase you will need at least your BOTRK before you can even consider roaming. When the laning phase ends that means you don't need to go top anymore. Split pushing gives you (more often than not) two options. Top or Bot. Your objective is to draw people towards you or get inhibs. Last thing... YOU HAVE A TELEPORT. Being selfish is important... but losing a 4v5 and then the game is a bigger problem. No one will ever turn down your help friend <3

As said before I have Jax down to a science. His E is one of the best moves in the game. Its a simple move, dodges all auto attacks and reduces some AOE affects. At the time I'm writing this the CD for Jax's E is long. The main thing I can tell you is to just save it. Look at the champion your about to fight at ANY time of the game and figure out, "Are they going to deal a lot of damage from auto attacks? Or abilities?". If its auto attacks then don't be afraid to jump on them. You can beat them by simply counting... Use your E, count your auto attacks so your about to use your third (assuming your lvl 6 or greater) and use your W. If they are alive you can do another auto and Q after as a auto reset or even just BOTRK for the extra damage.

I quit league the day I made this guide, but here is "how to win lane as Jax"
Get to lvl 6 asap after following the guide I put by the "ability" tab. Once you hit 6 the game changes. Remember, YOU ARE IN CONTROL NOW. Your playing the champion that was nerfed by the league itself. You are the best champion in the game. After you tell yourself this, its time to win the lane. Get creative with the trades now. The basic trade is to auto attack 2 minions to fool the opponent as if your going for CS, and then follow up with a Q into 1 auto attack, then resetting the auto with your W and finishing with your E to either retreat or continue the trade. This trade deals A LOT of damage... Feel free to do it multiple times.

GL summoner <3
Ps.... Jax mains don't talk to their team... or enemies, its your game