(Just saying, you might not agree with how I play)

Jungling
At the start of the game, go to the buff closest to bot lane and ask for a leash.
Upgrade your Counter Strike ability first of all. Pop your Counter Strike as soon as it spawns and start AAing it.
Once it gets to 500-300hp Smite it.

Upgrade your Empower straight after.
As soon as you finish that buff, go to the other side's buff. no need for Smite, just pop your Counter Strike then Leap Strike. Go straight to Gromp and after, Wolves.
When jungling, go for the smaller mobs first and when you get low smite the bigger one.
For E.g at wolves, go for the smaller ones first then smite the large wolf.

Fighting
Jax is someone who can engage a fight very well.
He has simple combos.

Engage: Leap Strike / Counter Strike / Empower
The Engage is quite simple, usually you would want to come in from the side.
Hit your Leap Strike towards the enemy and as soon as you land, use Counter Strike & Empower.

Mid Fight: Counter Strike / Empower / Master-At-Arms
when you are in the middle of a fight, just pop up Counter Strike, Empower and if possible, Master-At-Arms.

Escape: Counter Strike / Stealth Ward / Leap Strike
If you are getting low start running and use Counter Strike wait for the stun rather than doing it straight away in case the enemy attacks you. Quickly place a Stealth Ward as far as you think is needed then Leap Strike onto the ward.

Jungle Clear: Counter Strike / Empower / Smite
Jungle clearing is quite easy. aim for the smaller enemies first, later going for the larger one activate your Counter Strike then, instantly after that Empower later smiting the larger enemy if/when you are low

Ganking
Jax is great with ganking, usually you need to come from behind to get more damage on the enemy.
Use the 'Engage' combo to get the upper hand and if you need to, use the 'Escape' combo to get a clean getaway.
You're going to want to extort their flash and follow them until tower range.

Pros:
High burst damage
Surprise attacks
Good chase tactics
Great in team fights

Cons:
Low Crowd Control
Only one escape
Cannot take on large crowds

Jax is great in team fights, he is usually the most damaging in the team and he can tank quite well.
You can also ks very well, but I don't suggest doing that.
If you are playing Jax you should always join team fights as you can help your team tremendously.