This guide was originally written in Katarina's comment page (http://leagueoflegends.wikia.com/wiki/Katarina_the_Sinister_Blade#comm-663583). So other people thought I should write this in a blog, so here it is, slightly corrected somewhere.
First of all, Kat is quite easy to learn, I'd say really easy, but very difficult to fully master, you need to know when to initiate, or you and the whole game is dead. If you don't learn that, you wont succeed, but you will after reading this.
AD Katarina is not viable. She was barely ok some time ago, but after her rework she's not. Hybrid Kat is really stupid, she used to stack back when Spell Vamp stacked, the build's useless for a long time. AP Kat is the only way to go.
Katarina is amazing at farming, as soon as you're lv8 you'll clear the minion wave with a Q W combo and some autoattacks on melee ones. DO NOT overpush your lane with Bouncing Blades, sometimes you have to, but that sometime is when your opponent is missing and you're about to gank another lane or push a turret.
Katarina's harass doesn't do a lot of damage in early levels, your harass should be focused at constantly tossing out Bouncing Blades, you don't necessarily need to target Q on the opponent, sometimes you can let it bounce through the minions and hit it. Another harass is Q E W. You should Shunpo as soon as you use Bouncing Blades, and then Sinister Steel will consume Q's mark and give you a movement speed buff to retreat. You should compare your and your opponent's strenghts, you can't always just jump into it's face.
Skills, skill maxing, skill usage:
Skill maxing order is Death Lotus > Bouncing Blades > Sinister Steel > Shunpo. Bouncing Blades is maxed for several reasons: it's a skill that does the most damage, it's ranged and it opens a way to your combo. Some people may ask, why max Sinister Steel before Shunpo? That's because of: 1. It helps your farming. 2. The movement speed boost is increased. 3. It does more damage than Shunpo in early game. 4. It's AoE, everyone loves AoE. Thanks for an anon who pointed out that W could be also good to max before Q against certain enemies. This is not recommended against champions like Fizz or Galio, that can usually beat you in melee range early game, but it could be good to max against squishy champions with skillshot CC, where you need to get in close to them often (e.x. Lux, Xerath). In early game I use E so I can W the enemy, or use it just for utility, for example, to shunpo to a minion, evade skillshots... You must always remember that Shunpo can be used on allies and WARDS, use it often. It also reduces damage dealt to you, it helps you from dying in teamfights, or to mitigate some of the skills, like Vlad's Hemoplague, or Karthus' Requiem.
As I said, use Shunpo to evade skillshots. That is why you counter champions like Lux or Xerath, as soon as they begin their combo, you Shunpo behind them (it always blinks behind the target) and dodge it. You can also Shunpo to a minion if you don't want to appear at the opponent's face.
Your ultimate... Your glorious ultimate. It's amazing in so many ways of you learn how to use it. That is the most difficult thing with Kat. WHEN TO INITIATE? My tip, always memorize all your opponents and what hard CC or silences they have to stop your ultimate. Do not pick Katarina if they have a The annoying stunning biatch in mid. Ever. You should evade champions like Bitchcrank, Riven, Janna, Alistar, Kennen, Diana. Always wait until they use their CC, then make sure they really used all of it, THEN initiate and rip faces. It is important that you don't show up to your opponents, it's because if clever enemies will see you, they will save the CC for you and shut you down, you should try to hide in the beggining of a teamfight.
You can never forget your passive, the thing that makes Kat so strong. Always abuse it, try to get as many assists as possible in teamfights to refresh all of your non-ultimate cooldowns and reduce Death Lotus cooldown. This is what usually seperates the ok Katarinas from professional Katarinas. Always use smartcasts with her, if you don't, learn it. You're making a shame for Katarina's name if you don't. It helps you so, so sososososososo, so so much.
MY NEW SKILL MAXING ORDER:
Now I always max Sinister Steel first. It helps at getting kills and ganking, because if you're going in for the kill, you're still going to use all the combo. The reason why I max Sinister Steel is it has a shorter cooldown than Bouncing Blades. It also gives more Movement Speed the more you max it. After maxing Sinister Steel I max Bouncing Blades and then Shunpo. Note that if you're losing the lane, consider still maxing Bouncing Blades, or maxing them both at the same time.
Flash and Ignite are the best options.
Flash: this skill is the most OP Summoner Spell ever, like everyone says. It's used to escape, to do a Flash + Shunpo combo (they'll never see it coming), to have better positioning for you ultimate, and pretty much everything else that involves movement.
Ignite : The most valuable Summoner Spell besides Flash for most mid champions (not for Karthus). It adds up that extra bit of damage to finish that pesky Ziggs with 20 health. It also refreshes your skills' cooldowns, so you could use Ignite under turret and then Shunpo away after 2 seconds.
Exhaust isn't good, as both of SS mentioned above are a must. Heal too. Cleanse maybe oookkk sometimes, if they have a team like Talon, Alistar, Bitchcrank, Kennen and Tristana I had recently. If they have all 5 enemies packed with lots of CC, maybe it's good. I won't even mention other spells.
It helps you a whole lot. Never forget it gives you 20% Cooldown Reduction. If your jungler gives it to you, take it. If they don't, ask them to give you the 3rd or the 4th blue.
Katarina can build quite some items and is versatile. However, she has the circle of the perfect items that synergizes with her amazingly. My build is: + -> -> -> -> -> -> -> -> / .
is a standart pick for an AP Carry. and for early sustain, damage and gold income. is unbelievably amazingly fantastic, it amplifies your burst by 30%, adds AP, much needed CDR... Learn to use the active. is standart. is the only item that gives you Attack Damage, and it should stay the only one. Also adds Lifesteal for sustain, and the glorious active (LEARN TO USE IT). It could be replaced with if you're doing really bad and have no gold to save up for Gunblade, but it's not as good. Zhonya's Hourglass adds a ton of Ability Power and a life saving active (LEARN TO USE IT).ZHONYA'S HOURGLASS' ACTIVE INTERRUPTS YOUR ULTIMATE, stick this into your head. Abyssal is build if you need Magic resist against AP Carries, whilst Void Staff to counter Magic Resist, the Scepter helps to do it too. If you don't need any Armor, you can replace the Hourglass with Abyssal Scepter.
Other situational items: :versus heavy CC teams or when having an extremely rough game. REPLACE Skarner, Urgot, Warwick, Malzahar). : if you're doing so well it hurts for you (should happen a lot). If you never die and rack up all the kills, consider taking it early.''''' WITH ''''' IN LATE GAME, it's much more worth it. : versus suppresses that keep bugging you (
MY CURRENT BUILD:
I start withand , and rush a for the faster gold income. I don't get any anymore, I figured out that when you're skilled enough to dodge enemy attacks you don't quite need the sustain, although it's still as good as it was. I instantly rush an . Why, you may ask? It really helps your early game and makes ganks or surviving ganks way easier. As you're melee, you need some more tankiness then the ranged mids, and Magic Resist is for that. Then I get a , , and as before.
Kat, especially when 3 of her skills are AoE. Death Lotus used to take 3 seconds, Shunpo had a shorter cooldown for more slows and I think Bouncing Blades flew slower, it's viable, but not as good as other items. It was core on the old Kat.is a waste of gold. It adds AP, yeah, and it adds HP, yeah, but the slow is not important with the new
People always point out Kat is not completely Ability Power based to maximize it's potential, plus without 40% CDR your cooldowns are not short enough to abuse the passive, and you even won't have enough time to autoattack most of the time. It's not unviable, but one of the least viable AP item options.too. If you buy it, you put the mana to waste, plus
21/9/0. Offense Tree: Summoner's Wrath for improved Ignite, Brute Force for better last hits, Sorcery, Arcane Knowledge, Havoc, Blast, Archmage and Executioner. Defense Tree: 3 points in Resistance, 1 point in Tough Skin, it helps you freeze the lane and tank the minions, so they don't come to your turret, and I had nowhere to put 1 point. Then 4 points in Durability, and Veteran's Scars.
I have 2 Rune pages and use one Rune page for all Casters. I use Magic Penetration Marks, Ability Power per level Seals, Ability Power per level Glyphs, and Ability Power Quintessences. I'm extremely aggressive, so I have this kind of setup. Health per level Seals, Magic Resist Glyphs, Magic Resist per level Glyphs and Movement Speed Quintessences are also good.
Thanks for reading my guide, I know it takes a lot of space, but in my opinion, the wiki should have guides in it too, so players could get to know with champions. I wish I could write how to counter each individual mid champion, but the guide already takes lots of space and I haven't played against every single mid champion, it's like they say, if you do something, do it well.
Pictures, Videos, Jungling section, counter section, everything else you want.