Jinx's passive is called Get Excited, and what it does, is give Jinx extra movement speed every time she gets a kill, or assist, this is also activated when you destroy a turret. This is good for catching up to targets to get more kills, or a nice way to get out of a battle once you get a kill.
Jinx's Q is called Switcheroo, which is and interesting toggle which switches her basic attack to be a Cannon named Fishbones, or a minigun named Pow- Pow. Fishbones deals some area damage per hit, which can also crit. The Minigun is the top most gun in the picture, while the Rocket Launcher is the bottom most gun.
Jinx's W is called Zap, where charges up an electric shot with long range, it also slows. Because it take a second or two to charge it up, it might be easily avoidable, this is a move that you can use on a running away target, or vice versa to slow them down to either get a kill, or get away.
Jinx's E is called Flame Chompers. Jinx throws out 3 grenades in a line blowing up on a timer, but if an enemy champion steps on one, they get snared and take damage. This is good skill to use to initiate, or to get away, but take note that the grenades take a second to set up, so set them up in front of your target, depending on their movements.
Jinx's Ultimate, or R is called Super Mega Death Rocket, which is a global ultimate that deals damage to the first enemy champion it hits, then does some damage in an area around its explosion. This ult is mostly used as an execution, to get away enemies that think they get away, or just to steal kills from your allies around the map.
I would build Jinx the same way I build every ADC, building Berserker Greaves, Infinity Edge, Bloodthirster, Phantom Dancer, Runnan's Hurricane, and maybe Last Whisper.
Jinx's lore tells the story of, well, Jinx. It says that Jinx loves nothing more than having fun, which to her is blowing things up, like the city-state of Piltover. Jinx could care less about consequences, and thinks the whole police force is a joke, she uses them as her play things to have that much more fun, and her favorite toy is Vi.
While Piltover was once the city of progress, Jinx is seriously messing that up, destroying buildings of all kind, some would say Jinx is anti-productive, while I like to thing of her as..... misunderstood. That doesn't mean shes innocent though, things that Jinx has done has put Piltover, and its residence into a state of decline, causing the worst crime spree in the city's history.
Most of Jinx's intentions are a mystery as well as her origins, people say her weapons seem like hextech weaponry, which is equal to a steampunk style of things, like champions themselves or items, or pretty much anything.
Caitlyn then noticed the danger of Jinx and what consequences it could have if she wasn't stopped, so she started a man-hunt, or rather a women-hunt for Jinx. Jinx saw this as an opportunity to piss her and Vi off even more. Jinx marked the Piltover treasury with a picture of Vi's face, with a place and time shes going to rob/destroy the most secure building in piltover. Vi accepted the challenge and met jinx at that time. When they both showed up they started fighting, while Vi was breaking walls down chasing Jinx, she was helping with a bit of explosions, when Vi finally cornered Jinx, Jinx shot some rockets into the building with them both inside, tearing it down.
When Vi got out of the rubble, She noticed that none of the money from the treasury they just destroyed was taken, then she looked up and saw a message, the lights of the city read out " You'll Never Catch Me!"
"Oh look- I'm opening my box of care! Oh wait- it's empty!"
-Jinx
