Katarina is a mana-free, energy-free, hp-cost free assassin hero who specializes in poking, laning attrition, chasing, and dishing out tons of damage in team fights as long as she doesn't get stunned or silenced.
Innate Ability - Voracity
---- This is what makes Kat Kat. It is her Soul.
---- Whenever she scores a kill or assist, all of her cooldowns are reduced by ~13 seconds.
Q - Bouncing Blade
---- scales with AD - 100%
---- hits 1 + 1 x level targets, dealing 10% less damage each bounce
W - Killer Instincts
---- Gives Killer Instincts buff for 10 seconds
---- While KI buff is active using a Q or E skill will remove it and,
-------- next Bouncing Blade will do full damage on bounces and give 50% regeneration debuff
-------- next Shunpo will give Kat 10-25% damage reduction for ~3-4 seconds
E - Shunpo
---- scales with AP - 75%
---- targets friendly and enemy
---- deals damage if enemy and teleports Kat to the target
R - Death Lotus
---- scales with AD (33% of AD from items only) or AP (50%)
---- hits 1 x level targets repeatedly over a 2 second interval
---- try not to get silenced!
Generally accepted playstyles
There are two ways to play Kat, AD and AP, and people love to argue over which is better. Personally I think AD is better, so its what I'll write about because I know it best.
I'm sure all different kinds summoner spells are good on kat (heal, cleanse, flash, ghost, teleport, etc) but in my opinion exhaust and ignite are the best pair. They help you get kills early on in lane and push your attrition game as hard as possible, so much more than any other pair.
Regardless of whether you are building AP or AD, you don't want to just stack that one stat. I've found that Katarina needs both defensive and offensive items to work her magic. In particular, it's great to get some extra HP, a Banshee's Veil in most situations, and possibly a Guardian Angel.
Although the mana buff from banshee's veil is totally wasted, the protection from CC it provides can be more than worth it. It allows you to Q warwick with impunity and not have to worry about his ult. It allows you to dive into a team fight next to that sion that has been saving his stun for you and come out with a triple kill. It comes up and saves your ass when you least expect it.
The argument for AP kat is that there are great AP items like Rylais Crystal Scepter and Haunting Guise that boost defense and offense at the same time which have no parallel on the AD side. Although having lots of HP and having your ult slow is utterly fabulous, I still think AD is better. It's more well rounded and allows you to win the game without focusing on just killing people and without getting super lucky with voractiy-fueled double-ult quadra kills, That is, AD farms really well and takes down turrets really well, plus it allows you to be useful in team fights while your ult is down.
So without further blathering, here's how I play Katarina.
I'm not high elo at all just fyi
Or rather, how I don't play her. I will not play Kat if I think that my team has a lot less stuns / slows / silences than the other team. AD kat brings zero CC to the table, and is only really effective when she can float along on other peoples stuns, roots, and such. But when she does.....
My runes are magic pen marks, armor seals, cdr glyphs, and flat hp quints.
I go 9/21/0 in masteries to get the extra magic pen, extra damage reduction and hp.
I pick Exhaust and Ignite and a Doran's Shield plus Health Pot, and go mid unless someone else really wants to.
I level up Q, E, Q, W, Q, R Q, Q, E etc
Kat is a good mid champion. She can use her Q to either farm minions from a safe distance, harass, or both. In combination with her W, it gives great attrition. With the correct timing, you can actually out-attrition a Vlad who bought more potions than you did. Don't hesitate to use ignite to nullify a potion if Killer Instincts and Bouncing Blade aren't up.
If your opponent has 5 HP bars or less and no flash, you may be able to kill them or force them back at level 2. Try to land a couple Q hits beforehand, don't push towards their tower to bring them out farther, then if they take the bait, shunpo, auto attack, exhaust / ignite, Bouncing Blade, and keep auto attacking. If they get to their turret with < 150 hp, you might be able to do one final shunpo + BB combo for the kill, then use your refreshed cooldown on shunpo to get out before the turret kills you. Do not attempt this if you aren't sure or if there are no minions to jump to. Watch out for heals / red pots and stuns when you are tower diving.
If you didn't die or got a couple kills, you should be able to go back and buy a BF sword and boots on your first trip back. Thats an optimal situation. It doesn't always happen, but when it does and you are the highest level character on the map at level 8 or so, it makes you god. You hit hard with your Q and basic attacks, you are very mobile, and your ult can squish a health bar down to size in the blink of an eye. So, be bold, farm hard and harass, but don't do anything stupid. Constantly remind yourself that even though you don't have any snowball items, you are still snowballing. Kats dominance depends on her being bigger and stronger than everyone else. You want to have more hp than their tank during the early stages of the game, and you do it by starting out hp heavy (defensive tree, hp quints, dorans shield) and keeping it up by leveling up consistently. I usually buy a ruby crystal after my first BF sword if its practical and because Banshees veil is almost always a good idea. If they have a lot of casters I might even buy the negatron cloak early as well.
I view Kat as a carry. She should get the imba black shield on her. She should get healed. She doesn't engage first and doesn't get CC'd, because she has Banshees and waits for the enemy team to burn most of thier CC. If you start out in mid or a 1v2 lane and don't die, even if you don't get many kills, your Q and E should allow you to outfarm your opponent and be the highest level character on the map, with the most farm and damage output. You try to lever that and never let it go, just exploit it more and more.
Getting the red buff helps tons at any stage of the game. Blue is ok too for the CDR, but someone else needs it more than you in almost every case.
The final stage of your domination consists of two Bloodthirsters, a zeal / phantom dancer, Banshees Veil, and a Guardian Angel. If you never die thanks to your nice movespeed, lifesteal and defensive items, your bloodthirsters will become superbaddass and make your q, ult, and autoattacks rape face even harder. 1 extra bf sword or a GA is a tradeoff you can make, but either way you should be a hell of a pain in the ass for the other team, especially if you get a zeal / phantom dancer. The extra speed lets you chase better and makes your auto attack relatively ferocious. You can never disregard your autoattack when playing AD Kat.