Katarina is an assassin/carry/dps melee champion that excels at killing individual and multiple opponents quickly and at assassinating and escaping. She is somewhat squishy, but this is addressed with Killer Instincts + Shunpo, her masteries, her runes, and her items. He has no crowd control, so the best way to play her is to either wait and jump an individual champion, drops all her abilities, try to kill them as fast as possible and shunpo away or avoid aggro and jump into a team and drop her ultimate, picking off the weaker champions quickly and refreshing her cooldowns (note her passive). Katarina is actually not that hard to play and her abilities are straightforward, so she is a viable option for a new player. Note, however that she takes a while to master.

The reason why I build Katarina AP is because it just does more damage. That's it. All her abilities benefit from AP, whereas AD builds only benefit her ult and her knife, leaving Shunpo useless in terms of damage. Shunpo, however, has higher damage growth than her knife, made even higher with an AP build, and has longer range, making it a better nuke and harassing tool than her knife. Another thing to note is that the knife still benefits from AP and it will be about as strong endgame in this build as it would be with an AD build. In layman's terms, AD builds basically cripples one of her abilities and has a weaker ultimate than an AP build by far. Since you will find that most of your damage in team fights comes from your ultimate, it is clear why AP is better. Also note that you will reach over twice the AP than you would AD with the same amount of gold. This means stronger abilities and way more damage. Another reason is because all her abilities are magic damage and she has extremely short cooldowns, made even shorter with masteries, runes, items, and her passive. End game, Kat is able to go almost entire team fights without auto-attacking, making AD nigh-useless.

This build is quite similar to other AP Kat builds you might find on this site and it is not without reason. I have found the top rated AP builds to be very effective, but with my experience playing as Katarina, I think I have found a way to make them more efficient and stronger. One of the key points of this build is the ability to reach max cooldown with minimal effort and gold. With her runes, masteries, and items, she barely reaches 40% CDR at level 18, meaning no waste. This allows her to spam her abilities almost non-stop, which is very important for an AP character since her auto-attacks will do minimal damage. We also pick up some spell vamp and hp on the way, increasing her survivability when using her. The reasons for this will be described in more detail later on. This build is also quite flexible and I have included alternatives items, runes, and summoner spells in the text below based on the conditions. Being able to adapt to situations is a key to winning. For example, although this guide was built with 3v3 in mind, swapping out exhaust for teleport is just about all you'll need to switch to a 5v5 setting. Kat is flexible in this sense since her build for 3v3 and her build for 5v5 are almost identical.

Pardon the TL;DR. This is my first build on this site, so please forgive any errors and the simplistic nature. I have quite a bit of experience playing as Katarina, so please don't hesitate to give this build a shot. If you don't like it, think I should change something, or spot an error, please say so in the comment section. Thanks, and happy LoLing. :)

Pros

Amazing DPS
Great assassin
Great chaser and runner with Shunpo
Can score multiple kills with ultimate
Easy to use, straightforward abilities

Cons

Squishy
No CC
Useless if silinced/stunned
Not very good farmer

For Marks we get Greater Marks of Insight. The magic penetration is really the only useful stat provided to an AP Kat in terms of marks. The magic penetration will also increase your damage output since all her abilities are magic damage and since we get little magic penetration from other sources in this build.

For Seals we get Greater Seals of Evasion. The only other real vialbe option would possible be HP seals, but since we are not taking HP quints, I don't feel the difference is significant enough to be worth it. Evasion also is just as useful at level 18 than it is at level 1, whereas the extra 50 hp from HP Seals with make almost no impact at level 18. The Seals of Evasion also have pretty good synergy with your defensive masteries.

For Glyphs I get Greater Glyphs of Force. Cooldown would be the other viable option, but since we already have all we need from our Quints, AP is the best choice. Flat AP Glyphs also work if you like hitting harder early game, but I feel the 25 extra at 18 helps quite a bit.

For Quintessences, we get Greater Quintessences of Celerity, giving us 7% CDR at 18, giving us a grand total of 40% at 18 with items and masteries. Self-explanatory.

These are basically standard masteries for an AP Katarina build. The utility tree is out of the question since many of them are mana based and Katarina doesn't use mana. I pick up Improved Exhaust, AP, CDR, and Magic Penetration in Offense and all relevant masteries in the Defense tree. This makes Kat more bulky, which helps her survive when using her ult, and the AP boost from Ardor is also a nice plus. In this way, no masteries are wasted.

I run Flash and Exhaust. Flash is beautiful with Katarina because it allows her to Flash then Shunpo away if there are no targets in range or Flash through a wall, then Shunpo to a minions for a sure-fire escape. It also allows her to get within Shunpo range of an opponent when chasing, letting her Rylai's slow and damage do the rest. Exhaust is useful for both chasing and escaping like Flash and it is very useful, especially before you have your Rylai's in keeping opponents within range of your ult. I've seen many Kats run Exhaust and Ignite, but Ignite is somewhat redundant on Katarina since you can easily finish people off with her low cooldown Shunpo or Bouncing Blade, and a buffed Bouncing Blade already provides reduced healing. Not to mention the escape potential of Flash and Shunpo is simply too much to give up. In 5v5s I usually run Flash and Teleport since Kat's really good in team fights with her ult and teleport lets her be there when they go down.

I get Bouncing Blade early on as a early harassment/gank tool. Shunpo is stronger, but as I have already mentioned, it is too risky, especially if you are early ganking to use early on. After that I drop 2 points in Shunpo so I have two damage dealing abilities. At level 4, it doesn't let me get another point in Shunpo, so I drop another point in Bouncing Blade. At level 5, I get one point in Killer Instincts so I can use it in unison with my ult and Shunpo at level 6 to do heavy damage and live. After that, I max my Shunpo at level 9, drop a point in Bouncing Blade at 10, upgrade ult to 2 at 11, max Bouncing Blade at 14, and drop the rest of the points in Killer Instincts.

I have tried many different starting items with Katarina and I have to say that Boots of Mobility and 3 HP Potions are the way to go. AP is kind of useless in the beginning since Shunpo is too risky of a harassment tool early game, especially with low movement speed (hence why I get Bouncing Blade at level 1) and Bouncing Blade benefits little from AP. Defensive items help somewhat, but even with them, Katarina may end up taking to much damage in the start due to her low attack damage and her close range. Because of this, it is important that she is able to move in and out of range, and harass with Shunpo, her main damage ability outside her ultimate and move back quick enough to not take much damage. The boots allow her to do this. Due to her short range (she may take a bit of harassment early game) and her low farming ability, it is essential to get HP Potions to keep her in lane long enough to get her next item. Next we rush Mejia's Soulstealer to get maximum effect out of it. If the enemy is hugging turret or if you are having trouble getting money, pick up your upgraded boots first instead. Mejai's is good because it is easy to have it pay for itself when playing Kat and the AP really help and because it lets us reach 40% CDR with very little gold. If you are having trouble getting kills early and/or are dying, skip Mejai's altogether and rush Hextech revolver. After Mejai's, we get boots. If you are up against a lot of CC, get Mercury Treads. Otherwise, get Ionian Boots of Lucidity for the CDR. If you skipped Mejai's or you're up against Magic Resist tanks, get Sorcerer's shoes instead. Next, we get Hextech Revolver. I've noticed that most AP builds exclude spell vamp and I can't see why. This item costs only 1200 but it allows Kat to lane MUCH longer and stay alive and even regain hp when using her ultimate against multiple targets. With this, it is possible to score kills and live even with low hp. Next we get Rylai's for the HP (survivability) and the spell slow effect, which really helps her chase down opponents and keep enemy's in range of her ultimate. After that, just build a death cap. You will most likely have enough survivability from Rylai's and Hextech Revolver to where all you need to do late game is just build AP. Also, with Hextech Revolver, you gain survivability when you gain AP. After your Deathcap, you may want to upgrade you Hextech Revolver into a Will of the Ancients for the Added AP and spell vamp. The aura will also benefit allied casters in team fights. If the game is still going, get a void staff. The passive will shred through those annoying tanks and give you the magic penetration you were lacking before. If you bought Sorcerer's shoes earlier, you may not need the magic penetration and may want to get a Zhonya's Hourglass instead, more CDR, or just another Deathcap if you want to stack AP like a boss. The last item is mostly situational.

Harass with Knife. Don't go after kills unless you're confident. Focus on killing minions and use movement speed to avoid being harassed. Don't hesitate to use HP potions early since they won't be as useful later on, anyway. Try to keep laning until you've run out of potions, are low hp, AND have enough money for you next item. Kat isn't very good at farming, so it is important to land as long as possible. It you are laning against a ranged squishy, try to harass with Shunpo and bring the fight to them. If you up against someone more bulky, focus more on farming. Try to save your summoner spells until you reach level 6. Once you're level 6, try to harass down your opponent to around half and make sure all you abilities are off cooldown. Activate killer instincts, throw your knife, Shunpo to them, exhaust, then ultimate. If they aren't dead after this and if they try to ghost/flash away, follow them with you flash and Shunpo, which should be off cooldown. You will use this skill sequence in just about every fight. You do not want to bring down people's hp with your ultimate, you want to either kill them or bring them to low enough hp to where a Shunpo will kill them. This is because you are stuck in one place when using your ult, so if that doesn't kill them and they won't be dead from a Shunpo, you just lost an opportunity for a kill. Your ult does enough damage to where it can usually kill champions when they still have enough health to be confident enough to lane or even get close to you. You must use their confidence to your advantage.

There isn't much to say for this one. Try to farm as much as you can before ganks start happening. Try to let a teammate initiate/take aggro and then jump then with your ult when they commit to the kill especially if they are low hp. You never want attention to be on you before you use your ultimate because you will most likely get CCed and killed. If you are in a 5v5 and level 6, use your teleport to your advantage and try to gank when your ultimate is off cooldown. This is because Katarina sucks at farming and benefits more from kills, especially with Mejai's Soulstealer. As a general rule, do not initiate if you do not have you ultimate ready unless you are fed or high level because that is your main source of damage. If you have Hextech Revolver and you are low and someone half of less hp commits to you or turret jumps you, do not be afraid to jump them with Killer Instincts + Shunpo + Ultimate because it does a lot of damage and the spell vamp and damage reduction may keep you alive better than running would. Always remember that your job in the team, especially late game, is to be the assassin, not the pusher. Let you teammates do most of the pushing while you maintain high hp to kill opponents in team fights or ones stupid enough to commit to a kill when you're around. That's about it. Good luck, and have fun!