How to play Jinx on LoL
Let's start showcasing her habilities.
Her passive gives you a movement speed and attack speed increase when you kill an enemy or destroy an enemy turret. You can try to use this to your advantage when you want to chase someone down or escape. Sometimes you'll find yourself hitting a turret, and when it's almost down an enemy appears. Most of the time it's best to keep hitting the turret so you can activate your passive and run away faster.
Her Q, when in the Pow-Pow form, gives you an increase in attack speed with every hit, and you can stack it up to three times. It's best to leave it in this form when hitting targets with close range or hitting turrets and neutral monsters, because the dps (damage per second) will be bigger. When active, the Q is named "Fishbones".
Her W is a shock gun. It has a very large range and slows the enemy down. It's useful when you're chasing someone and you want to slow them down to get close to them, and also to check bushes for enemys. It has a delay to be cast, and it's not difficult to side-step, so choose wisely when to use it.
Her E is a trap. The trap roots the enemy for a few seconds, and then explodes, dealing damage. The trap also doesn't activate right away, so you have to wait a little for it to set. You can use this ability when you're fighting with someone, so you can get a few precious seconds of free hit, or you can save it for a moment where you want to escape.
Her ultimate is a giant missil that can cross the entire map. The damage of the ult increases with the distance that it travels and with the ammount of life the enemy it hits has lost. So, the lowest the life of the enemy, it causes more damage. It also deals splash damage, so it can damage enemies nearby. the ult doesnt hit neutral and epic monsters (as dragon and baron), but they get damage from the splash damage. You can try to steal a dragon or baron like that (I already did it, so it's not impossible).
I hope this guide helped a little.