Not Updated For Current Season
Well, I decided to make a guide for Katarina for those of you who are beginning to get to grips with her and want to become more advanced with her.
Why build her with AP?
I believe that if you're new to Katarina you should start with the AD builds, since her ulti can benefit from it faster than the AP, also you can play more safely by just tossing Bouncing Blades towards enemy champs by your tower and do somewhat decent damage to squishes.
However, if your play style with Katarina is improving and you know when to Shunpo in and unleash all hell to the enemy team and survive then you would definitely benefit more from the AP builds. Since this is more of a 'Mid Elo' guide I wont be going on to tell you all the mathematical advantages you get from each build, so you'll just have to trust me ;)
The first items I recommend are Dorans Shield and a Health Potion
These items are good for survivability to stay in the laning phase to hit lvl 6 fast. Now you have these items try to get mid, some people may question you and laugh but they wont be once you are ganking both lanes and picking up the kills (and giving them the assists ;P).
How to play mid is by keep using the same combo of: Killer Instincts > Shunpo > Bouncing Blade .
The Killer Instincts gives you a small but nice damage reduction for when you Shunpo into your enemy and the Bouncing Blade is to just give them that extra bit of damage before running off to hide behind your minions again. If you keep doing this until the enemy champs life is below half, you could go for the kill and exhaust/ignite them. The gold from this should let you return to the shop for your shopping spree.
The next item you want is Mejai's Soulstealer and Boots of Speed , so try not to leave until you have 1585g. If for some reason you need to go back to the shop before you have that much get your Boots of Speed and an Amplifying Tome if you can. By the time you have Soul Stealer you should be around level 6. This is the time you want to co-ordinate with your team to get in those early ganks and build your stacks. Try to aim for the squishes when ganking, a good tip for ulting in these kind of ganks is to Shunpo into them then walk a bit behind them and ulti, this either makes them run towards your allies or run all the way through your ulti and leave them in range of your Shunpo (if their not already dead). It's always best to initate a gank/fight using the following abilities in this order:
Killer Instincts > Shunpo > Exhaust > Death Lotus > Bouncing Blade / Ignite . This should kill atleast one enemy which will leave your Shunpo almost ready to be used again to finish off the ones that managed to survive (aslong as you know you wont die).
Now you want to work towards completing your Boots. The decision on what boots to get is entirely on what the other team has, so if they have got a lot of disables get the Mercury Treads Mercury Treads, if they have a few auto attackers (Yi, Tryn, Jax etc) go for the , you'll be surprised at how the dodge can save your life, if they dont really have either and you're carrying the team with loads of epic ganks, go for Boots of Mobility . Sorcerers Boots are an option too if they are stacking the magic resist.
After your Boots comes Zhonya's Ring . Good ol' Zhon's will be your main source of survivability during team fights for when you're getting focused, while giving you a huge boost in AP. Make sure you put Zhons in a good item slot that is easy for you to press, for example item slot 1, since you wont always have a big time span to use this item you want to know what button to press to activate it instantly. Dont waste this items Unique by using it randomly in team fights, only use it when you're getting focused.
After you have Zhonya's get Guardian Angel . This will help you maintain your stacks while giving you a nice boost in Armour and Magic Resist. Just having the glow of Guardian Angel around you can be enough to scare people from focusing you. Which is a BIG mistake for them. However if you are focused, it will take them too long to kill you, because your team should be able to handle them all by the time you've finished using your Zhon's and have finally resurrected from the dead thanks to Guardian Angel, allowing you to help finish off the rest of the team and take the easy pickings.
I normally get into buying parts of Guardian Angel by the time the games finished, but if its not this is where you just want to just keep increasing your damage output. This is where your preference to how the games going comes into your decisions on items.
Sell the Doran's Shield if you're at the shop and need the 200g to finish building an item AFTER you have Guardian Angel, or if you just need the item slot free.
The next items you really want are:
Rylai's Sceptor Rylai's Scepter - This is a GREAT item and can be bought earlier than guardian angel if your doing amazing and have no risk of loosing your stacks. I normally buy this if I'm loosing too much health from AoE spells since it gives a good health boost and slow which can help you get away or keep them in your ulti's range for longer. The slow is activated by Bouncing Blade, Shunpo and your Ulti! And Rylai's also gives you a decent amount of AP.
Void Staff - This is good if the enemy is stacking their magic resist (A juicy -40% MR to enemies FTW) and also gives you some nice AP.
Abyssal Scepter - This is nice if those mages are dropping your health bar a bit too easy since it gives nice Magic Resist while giving you a nice AP boost. Also its passive Aura gives the enemy a nice -20% magic resist.
Lich Bane - I've never tried this, mainly because a game has never gone on long enough, but this could help make your Shunpo nuke into a double nuke, it's worth trying if you have the gold!
Items I don't recommend:
Nashors Tooth - I don't like this item mainly because it only gives a small amount of AP and it's CD reduction I feel isn't a big enough reason to buy this item. You should get a big enough CD reduction from your Mejai's and Golem Buff, and then leave the rest to your passive, Voracity .
Why these Runes?
Quints x3 Flat Health - These runes are good for giving you a nice bit of health which is what helps you to keep annoying the enemy and allows you to take that bit of auto attack damage while you run back to safety after a shunpo.
Marks x9 Magic Penetration - These help for you to penetrate some of their magic resist allowing you to hit that bit harder with Shunpo.
Seals x9 Dodge - Also help to dodge some auto attacks during your Shunpo aggro to the enemy.
Glyphs x9 Flat Ability Power - Lets you hit harder early on.
Why those Masteries?
Yes it might seem stupid to not receive that extra 5% melee and magic damage, or that 5% extra defence. But honestly you don't need either. The offensive mastery tree gives you enough AP without wasting mastery points on AD parts of that tree to get that +5% and as for the offensive your more of an in and out champ who shouldn't be in the battle long enough to take serious damage, and it wouldn't be worth sacrificing one point from the offensive tree to gain that 5% defence.
Why level abilities this way?
Well it's mainly down to your preference. If you cant aggro the team with Shunpo successfully then levelling Bouncing Blades may be better early on. But Shunpo is your second major damage output with this build, so it's quite necessary to level it as fast as you can.
The beauty about this build is you can be a bit more self-reliant as a carry and you are a bit more in-control of how the game goes since you're the one doing the ganking and have 'alright' survivability to do so without getting owned fast, allowing you and your team to get the gold and that's pretty important since Season 1. As we all know how normal and ranked solo games can go. :)
Now get out there and start carrying your team to victory!