Katarina is one of my favourite champions for many reasons.
Once Katarina hits level 6 you will immediately be able to take out a squishy opponent, she has amazing harassment abilities, and with no mana to worry about, you have no need to be conservative with your skill spamming.
She can be played in virtually any lane, including mid, and as the carry or support depending on the circumstances.

Videos:
24 minute game recorded with LoL Replay.

Note: Some videos are interactive, and therefore require LoL Replay to play.

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Pros: - Great Farmer - Very strong from level 6+ - Easy to harass with Bouncing Blade - Great mobility with Shunpo - Very hard to be ganked. - Voracity resets most cooldowns on kills - Has a Spin-To-Win

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Cons: - Very reliant on cooldowns - Rather weak solo until level 6 - Death Lotus can be interrupted very easily - Can be focused down easily - -

The first skill you get really depends on what you're doing first, for an early gank, Shunpo, a fight with a teammate, Shunpo, for anything else you really want Bouncing Blade first, so you have that long range harass, and so you can leash buffs for your team easier.

Preparation is a very useful skill, however it's much better to get the damage per skill level from Shunpo and Bouncing Blade, than it is for the very small ammount of damage reduction gained each level.

For Death Lotus, I think it goes without saying to get it at each possible level, this skill will be your main damage dealer, and should always be used with Q>W>E>R when possible, or W>E>R when necessary, as the damage reduction will often save your life while channeling.

Death Lotus's main weakness is the fact that it can be interrupted by any stuns, snares, fears, silences, knock ups, and knock backs, so you should always either wait/try to make them waste any interrupts before jumping in with Death Lotus, or wait for a teammate to initiate against a tankier champion.

Towards the end of Death Lotus, or if they are about to move out of range, it's best to cancel it early and try to lob a Bouncing Blade for the slow from Rylai's Crystal Scepter and some extra damage at a longer range, or in some cases even to finish them off for the kill.

Another fun thing about Katarina is her passive Voracity, whenever she gets a kill or assist her cooldown all get reset/reduced by 15 seconds, so don't be afraid to Shunpo under the turret for a kill, as long as it's a sure kill on a person without heals/heal summoner spell, and you have a minion near by to Shunpo back to when your cooldowns reset/get reduced by 15 seconds. Don't forget Preparation will also slightly reduce turret damage.

When you jump on an opponent with Shunpo, if you use Preparation then Bouncing Blade> Shunpo (W,Q>E[>R]) from far enough, you will get the effence of Preparation on Shunpo for the damage reduction instead of reduced heals from Bouncing Blade, making it much easier to harass without taking too much damage, or to give you a perfect chance to use Death Lotus.

Note: If you can do it fast enough, you can also do Bouncing Blade> Preparation> Shunpo (Q>W>E[>R]), and still get the reduced damage effect. (Yes, shunpo is that fast.)

Ok well for items I almost always start with a Regrowth Pendant, and Sorcerer's Shoes right after, this lets me dodge around any skill shots, harass well and get back out without taking too many hits, and the regen from Regrowth Pendant heals me up afterwards.

Next item I get is situational, but will be either Amplifying Tome or Blasting Wand depending on how well I'm doing and how much money I have, normally I would go Amplifying Tome + Giant's Belt for decent damage, and amazing early game HP, then building right into Rylai's Crystal Scepter for the slow.

Now I used to get Hextech Revolver before building right into a Rabadon's Deathcap, but after playing Katarina for a long time I've found that it's much better to get Hextech Gunblade earlier, giving you not only another skill, but also an amazing slow that can be used for offence OR defence, and this is on top of all the extra sustain you gain from it.

At this point you can go for either Zhonya's Hourglass or Rabadon's Deathcap, if the team your fighting is balenced, or even worse AD heavy, grabbing a Zhonya's Hourglass first can be a life saver, however, if you think you'll be ok without armor for a bit longer, you can always pick up a Rabadon's Deathcap first instead. Also, if you *really* need some magic resist, you can always grab an early Abyssal Mask.

Now were getting close to end game, so we probably want to pick up either:
An Abyssal Mask to give us that extra bit of magic resist and that reduce magic resist aura for ourselves and out team.
A Void Staff to help killing tanks and people with high magic resist, but leaving you a little bit more squishy.
And finally if you really just can't stay alive long enough, there's always the old Guardian Angel, which you can really take advantage of with Shunpo+ Flash to get away from the other team camping your dead body.

Why I chose what I chose:

Greater Quintessence of Ability Power, nice early game AP which you can add on to by having ignite mastered and on cooldown.

Greater Mark of Magic Penetration, hits nicely through low MR, a must to do the most damage you're capable of.

Greater Seal of Regeneration, a controversial choice but one I think fits Katarina's play style perfectly, and combined with Regrowth Pendant gives you amazing sustain. Since Shunpo always has her fighting close, these let you harass the opponent down, take a moment and regen your HP, then finish them off, or at least drive them out of lane.

Greater Glyph of Cooldown Reduction, reduced cooldown is essential for Katarina, the more cooldown she has, the more chance she has to win a fight, with these for early game and Sorcery + Enlightenment to carry through to late game, she has the perfect balance of early and late game cooldown reduction.

Greater Glyph of Scaling Cooldown Reduction, these are also an option, but you will be sacrificing a great deal of early game cooldown you won't get back until level 13.

| Arcane Knowledge with Summoner's Wrath , which is very important since we only have to sacrifice 1 AP point to take it, and we get 5 AP, as well as 5 AD back in return as long as ignite is on cooldown.

With the Defense tree we took as much armor and magic resistance as we could, followed by Vigor for health regen, instead of Durability and Veteran's Scars for flat health; The reason for this is, Vigor is the best mastery to augment our previous choices of the Regrowth Pendant and Greater Seal of Regeneration's.

Now with the points we saved by taking health regen, we can max both Indomitable and Evasion , and move on to one of the most important things for Katarina, cooldown.
The cooldown per level from Enlightenment will help you carry you through the late game fights, helping to accent the flat cooldown we got from the Greater Glyph of Cooldown Reduction's and Sorcery.

Finally we finish up with Honor Guard and Juggernaut , giving us reduced damage, extra health, and a little resistance to stuns, which just might help you live a little longer to get that extra kill, or to escape with your life after that triple kill.

Though the most common Summoner Spell for Katarina is Exhaust, I would always take Ignite since the extra skill/damage can make the difference of winning fights, vs getting shut down early, it also give you the extra 5 AD/AP from whenever it's on cooldown from Summoner's Wrath , which can be useful early game.

With Rylai's Crystal Scepter + Shunpo you will already have a 35% slow to go along with Death Lotus, making it so that unless they stun/interrupt you, they will almost always take full damage.
Once you finish the Hextech Gunblade you have yet another amazing slow, so you have very little need of carrying Exhaust.

Flash can be very useful for escapes, but with Shunpo you can also use it as a supprise attack, or a meens to finish off a weak opponent. Just poke at minions from far, wait for the opponent to move up a little, then Flash> Shunpo or Flash>Full Combo without Bouncing Blade to completely catch your opponent off guard.

Well that's it for now, this guide is all very new so I'll add a lot more details and options later, as well as make it look cleaner/better.

Thanks for reading my guide, please give it a shot, you might be surprised.
I hope you guys learn to love Katarina like I do.

Feel free to comment with any feedback or ideas.