Hi, I am heat200, I have mained Jinx since her release and I love playing her. I have always been perfecting my builds with her and the few other champs I play and I think after having used the mainly the same build I really would like to share it with the world! I currently have 560k Mastery Points on Jinx and hope that'll be credentials enough to get you to try the build out. (ALSO, I picked Mafia Jinx for the backdrop because being able to spam her jukebox is fun 😃 The other skins have nothing spammable unless you want to spam her head making a bobbing motion for her dance which looks weird and is only fun with the right type of friends :| xD)

Jinx is a STRONG but a pretty IMMOBILE champion as well. She has great synergy with this build and I will explain why I chose each item and found each one to be great on her.

Infinity Edge:
Infinity Edge is a great damage item and on Jinx crit works wonders. I rush it because of the damage boost it offers by itself. When compared to the other items in this build, this one ALONE give the greatest damage boost, hence why it should be bought first, to keep your advantage or to try to give you an advantage in damage as early as possible.

Runaans Hurricane AND Berserkers Greaves:
I would build this SIMULTANEOUSLY with your boots for 2 reasons: First, YOU NEED MORE MOVEMENT SPEED, Movement speed is so necessary on Jinx because if you don't have enough you can't escape or kite people with dashes or high speeds anywhere nearly as well. Second, Runaans is an essential item on Jinx because of the way it interacts with her rockets. It gives the greatest boost to her damage output if you've already build infinity edge since crits do more damage already, splash damage is nearly or about tripled to clumped targets AND the crit chance is now increased along with your attack speed for a total crit chance of 52%

Rapidfire Cannon:
This item gives Jinx great benefits. First, when this item is charged you outrange tower range, this is great for sieging 1st and 2nd Tier towers which do not heal, this item charges fastest by attacking a minion wave with your rockets if you already have Runaan's so instead of running back and forth like a fool, use a wave to replenish it's charge near instantly 😃. Second, This item give you move attack speed, greater crit chance and on top of the bonus range that becomes even more useful once this build is complete it gives you bonus damage on hit when fully charged.

Your 4th/5th items (Frozen Mallet/Black Cleaver):
These 2 items are interchangeable depending on how your game is going. BUT why are we building these 2 items rather than another armor pen item and a life steal item? Yes life steal is great and most cases it is usually very necessary but due to the synergy of the utility awarded by these two items we can skip out on the life steal items, specially considering our keystone is warlords bloodlust which in many many cases is enough, it doesn't sound like enough but it really is towards the end and here is why. The combination from the extra range from Rapidfire cannon when going into a teamfight PLUS the Runaans will allow us to hit 3 people before they get into range of us since the range increase also applies to runaans. Runaans will then proc frozen mallet and black cleaver allowing us to kite the people we've hit fairly easily and since they're champion we will pick up some health off of these 3 champions. The slow + speedup will let us either move around or back while keeping the enemy like a Xin away from dashing range. This is essentially free damage 😃, then once they're low they'll want to run away BUT 😃 YEAH 😃 They can't because the slow and speed will keep you in just enough range to keep hitting them and if they manage to get slightly out of range your Rapidfire cannon will kick in pretty soon and BAM they're slowed long enough for you to get right back in range for another 2 hits and this process will repeat again and again for most of the end game

Pros:
-You have greater opportunities for kills with this build unless they focus you down HARD
-You have greater ability to keep enemies just out of their ability to reach you which is great because if they manage to catch you you'd have serious trouble escaping
- You have about 2900 Health along with 100ish armor and 70-80 MR at the end of the game, not many people will be able to TRULY one shot you unless you get caught, and due to the utility provided by the set, it wont be too easy to get caught unless you misclick during a crucial moment or misclick more than you could've afforded

Cons:
-You'll have a serious marker over your head in the late game, ADC's generally already have a pretty big target on them but with this you'll get REALLY annoying to the enenmy
-You'll lack some extra defensive items, the utility itself is great defense but it's not very useful against certain type of champions (Nocturne, Akali if she has enough ulti stacks
, Leblanc Dash->Dash->BOOM, Zed W->R) for those champs building something like a GA would be more useful but if you help your team be more aware of the surroundings then you should be safe enough for most of the game
-Lack of Lifesteal, due to lack of lifesteal in the build the Elixir of Wrath becomes really necessary for the extra sustain, don't forget to keep buying them at he end of the game. If short on gold to buy one you should do some jungle camps to get the gold for it.
--Con of Jinx: Jinx requires some extra attention to details, warding is something of greater importance with her due to her lack of mobility, it's bad to get surprises with her because most surprises will end up in your death TREAD CAREFULLY.