So you chose to play the role of the stealthy assassin, ganking side lanes and assisting teammates along the way. But did you know that Katarina can turn the tides of any battle with an effective playstyle? I'll show you how to dominate mid lane in the early game, be a true assassin with constant ganking in mid game and become an unstoppable powerhouse in the late game. Let's look at her abilities.

Pros:

Very powerful burst damage with AP build.
Effective during all game stages.
No mana cost makes for exceptionally long laning.
Easy chase/escape with Shunpo.
Some farming abilities with Bouncing Blade.
Looks hot in the Red Card Katarina skin.

Cons:

Very squishy without Rylai's.
No mana cost means weaker abilities.
Killer instincts is on a very high cooldown.

Q - Bouncing Blade - Katarina throws a dagger that bounces from enemy to enemy dealing flat damage + bonus physical damage + bonus magic damage. Subsequent hits do 15% reduced damage. The number of enemies hit and the magic damage scales with skill level.
This is your main farming and harassing ability. You want to level this up as quickly as possible to get your main items fast. With practice you will learn the maximum distance for the blade to jump between two minions.

W - Killer Instincts - Passive: Katarina has increased damage on her physical attacks and abilities. Active: Katarina's next basic spell has a bonus effect. Bouncing Blade does full damage to every target hit and reduces incoming healing to the target, or after using Shunpo Katarina's has increased damage reduction.
This skill will take a couple of errors to master. Always use it when available. W- Q is a great farming combination, while W- E is what you need to get into the team fight. The damage reduction from it is amazing, it's like having 20 Leviathan Stacks with the push of a button. Never underestimate Killer Instincts. A good strategy while laning is to use W- Q to reduce the enemy's potion effectiveness by 50%.

E - Shunpo - Katarina instantly moves to her target location. If the target is an enemy, she deals magic damage.
Your main initiator/finisher/escape/chase tool. Shunpo is a great nuke, often forcing even Baron Nashor to focus you, instead of your teammates, because it does excelent damage and has about 0.8 AP scaling, perfect for your needs. You'll want to max it out after Bouncing Blade, because its damage starts to shine in mid game. A really powerful combination is Shunpo, followed by a hit, enhanced by Lich Bane. This combo alone takes about 50% HP off squishies.

R - Death Lotus - Katarina goes into a state of extreme focus, throwing daggers with unrivaled speed at up to three enemy champions. Channeled.

Katarina's deadliest and hardest to master ability. Because it channels for 3 seconds, you become extremely vulnerable. Moreover, Death Lotus does so much damage your practically highlighted for every stun/disable the enemy's got. Katarina's ultimate is best used in the middle of the fight, when the first stuns have been blown. Banshee's Veil helps immensely for your full damage output. Also, Death Lotus is on a very low cooldown, even lower with her passive. Use it whenever you can to land the kill.

(Passive) - Voracity - Every time a champion Katarina has damaged in the last 15 seconds dies, she gains 25 gold and her cooldowns are reduced by 15 seconds.
One of the best passive abilities in the game IMO. This will allow you to land double, triple kills or even ace an entire team. You need very quick reflexes to make it count, spamming your abilities. Killing a champion instantly refreshes all but your ultimate ability. The extra gold doesn't hurt, either.

Before I go anywhere further I like to remind you: NEVER initiate fights. This is an art that you will learn over the course of learning to play the role of an assassin. Katarina must choose her moment precisely and that is where most Katarina players fail. They'll get stunned and focus - fired. Because this deadly beauty unleashes a lot of damage extremely quickly she is one of the highest- value targets for the team, ESPECIALLY if you happen to have a lot of Soulstealer stacks.

For Summoner Spells I usually go Exhaust Ignite. These two spells are heavily offensive, because Katarina needs some early kills. Her farming abilities are limited only to her Q, so she relies on heavy assassination play. Ignite is a must, although Exhaust's effects wear off as soon as you get your first major item - Rylai's Crystal Scepter. So a combination of Ghost Ignite or Flash Ignite is also viable.

Katarina is best left solo. She's not a good 2v1 player so mid works best. She can get fed a lot earlier, get her Rylai and boots and start ganking ASAP.
Start with a Doran's Shield. In very rare cases you can get the Amplifying Tomes, if you're facing non - harassment heavy teams. This item, along with your runes will give you a hefty 38 AP to begin with, but be aware that you have no means of regenerating health. If you have opted to go for improved ignite, blow it on level 1, giving you another 10 AP boost for your harassment.
Play defensively until level 6. Make sure you've done some harassment to the enemy champion, as well as last- hitting with your Shunpo. A very key element here is: never harass with shunpo! Shunpo is a dangerous nuke that puts you among enemy minions and the champion. You can get beat up so badly using this you might even be forced to recall. Stick to leveling up your bouncing blade and hitting minions adjacent to the enemy champion, saving Killer Instincts for when the enemy uses a health potion.
If you have followed my steps so far you'll be near your tower, at almost full HP, while your opponent will likely have about 60% HP. Make sure you time your attack perfectly, keeping in mind both the enemy's summoner spells, as well as their hero's ability to escape. If you're facing the typical Ashe/Kog'Maw/Caitlyn/Tristana/Miss Fortune etc., go for it. Start hitting a minion very close to the enemy champion. He will retreat but will soon come back to harass you. Now he's in range of your shunpo. E-Q-R-Exhaust-Ignite. Cast your summoner spells while channeling your ultimate for an extremely fast burst. You can kill a 600 HP KogMaw in about 2 seconds with this combination.
Leave mid lane as soon as you have enough money for basic boots and Rylai's Scepter. This gives you immense ganking capabilities. Hide in the bush and tell your teammates to let the enemy push. When they're ready,get them to initiate, shunpo to the nearest champion and Death Lotus him to oblivion. This strategy gives me flawless rampage scores almost always.
As we approach late game, your opponents will have learned to hate you. If you have your Mejai you probably have about 10 stacks now and working towards 20. Here's a tip: never show yourself to the enemy team. Stick to the jungle and bushes, following your teammates as they go for the 5v5 showdown. Your opponents will think it's a 4v5 and thus blow all their stuns and disables before they see you. When this happens, shunpo to the nearest enemy champion, hit R and watch their team get aced in seconds. Don't waste any time with Katarina. As soon as a kill or assist has been completed and your ultimate has worn off. Shunpo- Blade to the nearest champion again, repeating the slaughter. With tactical playstyle you can get two ultimates in a single fight!
If your enemies are cautios and save a stun for you, get either QuickSilver Sash or Banshee's Veil.
So there you have it, the AP guide for Katarina by ChefoSLR. Now get out there and start leveling up those killstreaks, but don't test your luck, Summoner.