First off, thanks to wRAthoFVuLK and ilke994 because this guide is largely based off their guides. By largely based off of I mean there is a fair amount of copying involved, not only because their builds are really good, but also because there's very little variation when it comes to building Kat. So everyone should go upvote their guides because they're amazing. With that out of the way, let's get into your AP/OP Katarina.

WARNING: WHATEVER YOU DO, DO NOT BUILD THE SAME THING EVERY GAME. THERE WILL BE GAMES WHERE YOU WILL BUILD THE SAME THING AS THE LAST GAME YOU PLAYED, BUT ALWAYS BE AWARE OF WHAT THE OTHER TEAM IS DOING.

Pros

Amazing Harass with Shunpo and Bouncing Blade
Incredible farm after you get Bouncing blade
If you get fed with her passive there's no
stopping you
Great Ult
Really good damage/ability scaling

Cons
Somewhat squishy
Easily CC'd
Easily interrupted ult
Can get focused if you start to do well
You will get accused of KS'ing A LOT.

I personally think that this is a stupid argument. Some people complain that the ratios on Kat's ult, Death Lotus as well as her Bouncing Blades allow you to build a better AD or hybrid Kat than AP Kat. Keep in mind, that as AD or hybrid Kat, Bouncing Blade becomes your main damage dealing spell. I prefer AP Kat for the following reasons.

1. In my opinion, hybrid champions of any kind are stupid. The AD/AP eventually balance themselves out on most champions, leaving you with basically the same damage as you would have if you had built all AD or all AP.

2. While AD Kat with Bouncing Blades will give you hella damage, you'll be neglecting your Shunpo which is THE most important skill that you have as Kat.

3. The cooldown on Bouncing Blade is longer than shunpo, and it does less base damage.

4. Your Shunporange is longer than Bouncing Blade, which enables you to burst much more effectively. With Bouncing Blade as your main damage spell, chasing becomes sloppy because you have to get in the shorter range of Bouncing Blade, then shunpo and wait for your cd's to reset. And trust me, you'll be doing a lot of chasing as Kat.

For more in-depth information on why AP is better than AD or hybrid, check out wRathoFVuLK's argument.

is a world-class passive. Early game it will help you get fed. Hard. Mid-game you'll be able to dominate team fights with the cooldown reduction as well as get fed with the bonus gold. Late game the combination of your early and mid-game and the cooldown reduction will destroy your opponents.

My skill order is pretty basic. In max order it goes Q>E>W>R. What that means is you max bouncing blade first, then Shunpo, then Killer Instincts, and of course you grab your ult whenever you can. While Killer Instincts is a great spell, you really don't need it until later levels because you should be focusing on farming much more than combat at the early levels. The important thing to focus on here is, BOUNCING BLADE BEFORE SHUNPO.. This is hugely important. Kat is a heavily item based champion. As such, farming is very important for her. If you max Shunpo first, I can pretty much guarantee you that you will have a mediocre-terrible game. Here's why: Katarina sucks at farming. Her slow base movement speed combined with her meh attack damage make her pretty terrible at farming without abilities. That's where Bouncing Blades comes in. Once you get past Level 2 or 3 of Bouncing Blade your farm becomes insane. Hang back behind your minion waves and let them damage the enemy creeps. Then, when you see several minions low, W then Q to make your Q do full damage on all enemies it hits and watch the gold coins fly. You simply can't do that by maxing Shunpo first.

Important Farming Information: In order to be successful in farming, you have to use your abilities. With practice you'll be a farming whirlwind. W>Q, then Shunpo to a weak minion to take it out. However, be careful when doing this with enemies in your lane. You don't want to get ganked while your shunpo is on cooldown or you'll be in serious trouble.

This build is pretty straightforward. The one explanation that it requires is selling your [ icon=Hextech Revolver size=50 ] after you get [ icon=Rabadon's Deathcap size=50 ]. The revolver is crucial early game. It gives you just the right amount of sustain with the lifesteal and then gives you an extra boost of ability power to get that edge of your opponent in the early game. Most guides will tell you to get a [ icon=Hextechgunblade size=50 ]. I always felt like this was never a good choice on Katarina. The ability power is nothing special, and the life steal really does nothing for her in the late game. So, if you prefer, you can get a Hextech Gunblade after your [ icon=rylai'scrystalscepter size=50 ] and eliminate the [ icon=AbyssalScepter size=50 ]. One of the nice things about AP carries is that there's often little to no variation in what they build. As Katarina is pretty much a glass cannon, there's not a ton of variability in what you should build as her. Of course there are always exceptions. If your opponents have some serious CC going for them and they just won't leave you alone to ult in peace you may want to consider getting an early [ icon=Banshee'sVeil size=50 ]. Likewise, if the enemy [ icon=Veigar size=50 ] is fed out of his mind and one of his [ icon=Balefulstrike size=50 ]s takes away half of your health you should get an early [ icon=Abyssalscepter size=50 ] and focus him down so that he's not as much as a threat. The same goes for [ icon=zhonya'shourglass size=50 ] if an enemy AD carry is harassing you.

This section and the farming section go hand in hand. While farming with you W and Q, target creeps near your enemies in order for the blade to bounce onto them and deal full damage. As someone who has played against numerous Katarinas, I can say that this is legitimately one of the most annoying things to deal with possibly in the entire game. Not only does your [ icon=Bouncingblade size=50 ] with [ icon=KillerInstincts size=50 ] deal full damage to every target it hits, it also applies a 50% healing reduction. So basically you can poke your opponent and farm at the same time. It's pretty amazing, and one of the things that makes Katarina great, especially in lanes against supports like [ icon=Soraka size=50 ]or [ icon=Sona size=50 ]. While harassing with [ icon=Shunpo size=50 ] can be extremely rewarding, it can also be really risky against champions with stuns or slows like [ icon=Taric size=50 ] or [ icon=Alistar size=50 ]. In short, you should only use [ icon=Shunpo size=50 ]for harass if your opponent does not have a disable and/or you know you can tank the damage involved if they turn around and smack your face.

[ icon=Flash size=50 ]:
An absolute must on Katarina. The Flash then Shunpo combo can be incredibly lethal for ganks or chasing. Non-negotiable summoner spell.

[ icon=Exhaust size=50 ]

A good summoner spell if you want someone to stick around just a little bit longer while you ult the **** out of them. Also good for making gaps even larger by exhausting the chasing enemy champion and then shunpo'ing to a nearby minion/champion. Overall a great Summoner Spell on Katarina. It's really a matter of opinion between this and Ignite

[ icon=Ignite size=50 ]

My personal preference for summoner spell on Kat because it lets you get that last bit of damage that you can't quite deal when all of your spells are on cooldown and your opponent is getting away. Has contributed to countless kills for Katarina players everywhere. You can really accomplish the same thing with either Ignite or Exhaust on Katarina, so it's just a matter of your preference.

TL;DR:
Get Flash. Choose between Ignite or Exhaust based on your own judgment.

As an assassin, you basically have the worst job on your team. It's your job to wait until someone initiates a team fight, wait until that team fight gets to about the middle of its progression, then run in and ult after everyone has already used their various CC's. It takes a lot of practice. You're basically a janitor. It's your job to clean up champions that are low after you teammates have died in a team fight. Because of this you're going to be accused of KSing a lot. Don't let it get to you. It's just your role as a part of the team.

My first build, I know it's not great, but I hope it helps you make your opponents complain about how OP Katarina is. Enjoy!