You can use this guide as a quick overrun of the champion or as an extensive guide. I have over 170 games with jinx and a 65% win rate. In my opinion she is one of the most snowbally ADCs. Get a kill or two early and win the game for your team. Give up a kill or two early? Farm up and win game.
Q scales very hard with leveling it up, which is why you will want to max it first. The attack speed stat of the minigun, and the attack range of your rockets is why it is so strong. Get a point in it first, and max it by level 9. Some newer players make the mistake of maxing w first, although this is wrong. The reason why this is worse is because your w scales a lot harder with straight up AD then it does with leveling it up. Basically, just buying items will do more good then leveling it up. Get a point in this level 2 and max it by level 13. Get one point in Flame chompers level 3 but max it last. It provides a good way to cut off escape routes, disengage, and help distance yourself from the people wanting you dead.
Early game your focus should be on farming as best as possible. Farm using your mini-gun, and switch to rocket launchers if you want to poke the enemy. An easy way to get free poke on the enemy adc is to auto attack them when they are going to farm a minion. By doing this the enemy will either take a free auto attack from you and get the minion, which is worth. Or they will trade auto attacks with you and miss the minion, which is worth. If you get good enough where you will not miss a minion also it works to the best affect.
By the time you have Infinity edge, boots, and static shyv you will want to group with your team. You will be doing a lot of damage, and if your team can properly protect you you should have no problem snowballing team fights. You can also take out objectives very quickly using mini-gun, whether that be dragons, or turrets. Use ult to make picks whenever possible, even if you miss your ult has a low enough cooldown you can toss it out whenever the opportunity arises.
Your goal is to do as much damage as possible to anyone you can reach. Use your rockets when the enemy is grouped together, and as they begin to close down on you, use your mini-gun to kite them back. At this time in the game you can push towers extremely fast. Use this to your advantage if you win a fight.