Hello everyone. This is my guide for Katarina - The Sinister Blade.
Katarina is an assassin and melee. She's pretty easy to play with and you can make A LOT of kills with her. Sadly enough her defense is fairly low and her basic attack (physical damage) doesn't mean much. However, her amazing abilities compensate. That's why you absolutely have to focus completely on AP (ability power).
I don't know if you find this important, but I'm not ranked (yet). Nevertheless I do think I have more than enough experience with Katarina to make a guide.
I tried to make this guide noob-friendly. Sorry for the pros who think my explanation is too simple :p
I hope you like it!

Pros

+ Abilities cost nothing! (no mana & no health!)
+ Everytime you kill an enemy or assist in a kill, cooldowns are reduced with 15 seconds, which means Q, W & E have no cooldown!
+ Pretty easy to play with.
+ Great minion farmer.
+ Great escaping ability: Shunpo.

Cons

- Squishy.
- Bad in early game.
- Ultimate is easy to stop.

Flash: Katarina has only got 1 escaping ability: Shunpo. And you can't use it all the time. That's why you need flash. I suggest you save flash for escaping purposes mostly. When chasing an enemy for example; don't use flash if you're not 100% sure you're going to get the kill. You could get in trouble later when you're the one being chased.
Ignite: This choice is pretty obvious since you're playing with an assassin. It's a great spell to finish off an enemy champion when he/she's trying to run away.
I suggest you don't use this spell if your enemy still has a lot of health. Use your abilities first.

I absolutely think these two spells are the most useful for Katarina. I'm not going to recommend any other spells.

Boots is standard. After that I advise you to buy Hextech Revolver as fast as possible. It already gives you some AP + the spell vamp keeps you alive in early game. You're still too frail though, that's why I chose Rylai's Crystal Scepter as second item. This item gives you the health you need, ability power and the passive is very useful.
Katarina isn't very fast, so I continue with Sorcerer's Shoes which also gives you magic penetration. From here on you just have to continue focusing on ability power items. Rabadon's Deathcap, Guardian Angel, Deathfire Grasp and lastly complete Hextech Gunblade. Hextech Revolver is very important, but you should only upgrade it to Hextech Gunblade when you have more than enough gold and when you already have your other items. Note: I always use the same items until Deathfire Grasp. Sometimes I use Wooglet's Witchcap or Void Staff instead!
I also chose Guardian angel because Katarina often gets focused (if your enemies are smart). Once she's fed, she's unstoppable.

Passive: If you kill an enemy champion or assist a kill, your cooldowns are reduced by 15 seconds. This means your Q, W & E have no cooldown!
Bouncing Blades: You throw a dagger that bounces to nearby enemy champions, minions or pets. Take this ability first! You can easily farm minions with this ability and harass enemy champions. In early game: don't target a champion. Target a nearby minion. Your enemy will think you just want to kill a minion and will, therefore, stay where he/she is. This makes it possible to farm minions and lower you enemy champion's health.
I max Bouncing blades last, but you should take a second point more early if no fights break out and you're still harassing your enemies and farming minions.
sinister steel sinister steel: Katarina whirls daggers in a circle. If you hit an enemy, you gain extra movement speed for 1 second. I advise you to max this ability first. It's also great for farming minions.
Shunpo: This is an amazing ability. You flash to the enemy you targeted and deal some magic damage. I already gave a hint: this can be used as flash because you can also use this on allies (you can't wound your allies of course!). This is your only escaping tool, but it's a good one.
Death Lotus: Great ultimate. You throw daggers to 3 nearby enemies. BUT, this only works when your enemies are close enough. Know you range! Advantage: you don't have to aim.

It's all about ability power. So feel free to use every ability power rune you can use.
However, this is my favorite combination.
I also like to use Seals of Armor and Glyphs of Magic resist.
But I still prefer Seals of Scaling Health and Glyphs of Ability Power.
These Marks and Quints are definitely the best for Katarina.
The magic penetration makes it possible to deal more magic damage and the Quints give you almost 15 AP at level 1 already!

In this chapter you can find tips and tricks on how to play with Katarina.

Katarina has low defense.
Therefore, it is important that you play very defensive in the beginning of the game. Patience is the biggest problem for 100% of all summoners.
Start with Bouncing Blades and try to farm minions and harass your opponent (check 'Skill Sequence' for more information about how to play with bouncing blades).
Wait until you've got all your abilities before you start killing. Don't worry: you can make more than enough kills with Katarina.
These are my killing combinations (i don't talk about ignite here) once I've got all my abilities:
1: Shunpo -> Sinister Steel -> Bouncing Blades
2 (if the enemy has (too) much health): Shunpo -> Death Lotus -> sinister steel -> Bouncing Blades
1: Definitely start with Shunpo. This way you make sure you're close enough to deal damage (and you surprise your enemy!) + after using this item you gain 15% damage reduction for 3 seconds. Because you're close you follow with Sinister Steel, which gives you extra movement speed for in case your enemy starts running.
If you're winning, your enemy will probably start running from this moment on. That's why I kept Bouncing Blades for last. Now I can still get the kill by throwing the dagger.
If he/she's still alive you start chasing (with the extra movement speed from Sinister Steel) and you wait for the cooldown of Shunpo. As soon as the cooldown is over you can start all over again.
Important: Be careful if you use your ultimate. Know your enemies; if a nearby enemy has the ability to stun... you might waste your ultimate (and get killed). That's the disadvantage of Death Lotus.

Most of the time I play mid with Katarina (which is the best option), but it doesn't matter much to me. When I do play mid I wait until I got all my abilities and at least Hextech Revolver as item, before I start killing. Katarina is a great ganker. I go top and bot time after time to clean up the lane. You're the assassin so you do the killing. Katarina is a great champion to finish unfinished bussiness. Nonetheless, you should play fair. Don't steal kills. Be a gentleman.

This was my guide for Katarina - The Sinister Blade (currently 3150 IP).
I'm going to adjust and improve this guide a little bit more in the near future.
But this is it for now.
I hope it was useful for you.
I Tried to keep it short, because we don't like to read much, do we? ;)
If you still have questions: just ask.
If I said something completely wrong: please tell me.
Sorry if I made spelling or grammar mistakes. English isn't my mother tongue.
Thanks for reading and don't forget to rate :)

Greetings,
CohibaNr1.