Jinx is one of the strongest AD carrys of season 4. She is powerful at all phases of the game. Jinx's increased range helps her dominate in the early game, while her AoE and attack speed help her throughout the mid and late game. Her passive allows her to excel in teamfights.

Pros:
-700 Range that few adcs can match (http://i.imgur.com/1YjjNeh.png)
-Built in ranged Tiamat
-Team fight mobility that can lead to multiple kills
-Amazing push
-Global AoE ultimate
-an 130% attack speed steroid at lv9
-Flexibility between damage and safety with Switcheroo!

Cons:
-a very low base range without Switcheroo
-No escapes
-Low base attack speed
-Very susceptible to high mobility assassins such as Fizz

Very standard rune page, I used to do AD quints, but life steal is so useful for early trades and sustain. The only time I would use AD quints is when my support has a reliable heal.

Switcheroo!
You want to max this first, because of it's range, attack speed buff and AoE damage. With a lv5 Switcheroo, you have 700 range, the largest in the game outside of kogmaw and twitch, a 130% attack speed buff with 3 auto attacks(largest in the game), and . Other benefits are a massive attack speed boost and AoE damage.

Tip: When you switch to Fishbones, the attack speed bonus you get from pow-pow stays with you, so you can have a 130% attack speed boost while having a 700 ranged AoE auto attack. This bug hasnt actually been fixed as of patch 3.14.

Zap!
Use to last hit, poke, and to chase opponents. You don't want to max this first, because it's utility is incomparable to Switcheroo! You get some extra burst, but maxing Switcheroo gives you more consistent damage, which is what you want in an AD carry.

Flame Chompers!
A good Flame Chomper can completely prevent an opponent from engaging or escaping. You should lead the opponent movement route to give the chompers time to arm. This will take time to learn. You max this last, because all you get is a slight cooldown reduction from each point.

Super Mega Death Rocket!
You can use it for sniping other lanes, but I don't usually, because it is a pretty long cooldown at level 1 and you will lose a 2v2 without it. Of course max it when you get the chance, because it does really good damage.

Most of it is self explanatory. I see a lot of debate about Dangerous Game, but it helps a lot with chaining kills. I can remember multiple instances where I could chain kills and bait other champions with my deceptively low health. With Jinx's passive it is very easy to chain kills.

Flash
This is self-explanatory, flash is useful in getting away and chasing.

Barrier
You want this in almost every situation. Barrier helps you block a lot of damage and can be used to bait opponents.

Cleanse
This is only if bot lane or the enemy team has a lot cleansible CC. Get cleanse if the other team has strong cc champions such as fiddle, leona, ashe, etc.

Starting:
This is pretty much the start you want for any standard matchup. If the other team pulls out double bruiser bot, you may want to considier .

Early Laning:
Everything after Starting is situational depending on how much gold you have. If you happen to have 1550gp, then go for the BF sword, but usually you're going to have to buy the vamp scepter and boots first. If you're having a tough time and getting chunked then go for the extra dorans blade.

Mid Game:
You should complete the Bloodthirster and Berserker's Greaves first depending on which is more convenient when you are in shop. You want to buy Bloodthirster over I.E. because a lot of Jinx's damage comes fromabilities, so the extra attack damage is more important than crit. You buy shiv because it is better with a Bloodthirster than PD is. Shiv also lets Jinx one shot waves with Fishbones, the Rocket Launcher.

Last Items:
So the order of the last items are dependent on the game. If you die really fast then you probably want to invest in a survivability item first or Furor boots which will help you kite better. If you are getting in a lot of auto attacks, buy damage.

To determine which damage item to buy first, examine who you are attacking in team fights. Usually the tanks/ assasins. If they have armor, buy Last Whisper, if not, then build Infinity Edge.

To determine which survivability item to buy, you look at who is attacking you in fights. If the problem is burst, buy G.A. If most of the damage is physical, buy Randouins. If CC is the real problem, determine decide whether you can use quicksilver sash to counter it. Hint: Knock ups are not affected by QQS, but suppressions are.

Keep using Switcheroo! to switch between Pow-Pow, the Minigun and Fishbones, the Rocket Launcher. Pow-pow should be used to last hit and fishbones for poke. Remember that fishbones does AoE and you can use it to lasthit and harass at the same time if the enemy is close to the minions. Poke the enemy down with Fishbones and Zap! then look for an engage.

Jinx is a strong pusher, so you should push up lane and try to take turret. Don't hesitate to take jungle or roam mid.

As with any AD carry, stay with your team, always stay in the back, and kite. Save your Flame Chompers! to protect yourself, and use Pow-Pow, the Minigun and Fishbones, the Rocket Launcher accordingly. pow-pow does more damage, but Fishbones is a safer choice. Use your judgement to determine which is better in your situation.