This is just what I run and have had the most success with since it isnt too specialized.

The easiest way to get ahead with Jhin in the early game is using his 4th shot crit on the enemy adc or support. You farm up 2 cs and as you go to get the 3rd cs move in towards your enemy laner and take the 3rd cs and immediately pop your 4th shot on the closest enemy (DO NOT give them time to back up; if youre able to do this quickly you will overpower them early almost every time.)

Never make the mistake of getting way out of position to get the 4th shot on them though; if you cant get it safely just secure more cs with it and try again.

Once you have the 3 basic abilities youre ready to hit some huge combos in fights.

A great combo that almost almost seems to get a KO is to first hit your 3rd/4th shot on the enemy (I try not to engage with any less)and simultaneously throw your grenade and then pop deadly flourish to stun them while simultaneously throwing out captive audience on them or slightly behind the way they will run. This way if youre not able to hit a nuke on them they will be slowed when the stun ends and you can pickup the rest with either basics or your ulti.

TL;DR Do this as quickly as you can for easy nukes.
(3rd/4th Shot + Grenade) -> (Deadly Flourish Stun + Captive Audience Trap) -> Basic/Ulti