So uh. I'm lame and don't know how to split up the boxes like all the other cool guides do, so bear with me, this guide WILL be made better as time goes on :)

Pros:
Amazing Ganking potential. Her full combo can take someone down from 1000+ health if done right.
Easy farm and harrass with QWE+ Hextech Gunblade
Not as squishy as some people believe.
Kites everyone :3
Just got a rework~

Cons:
No more tentacle hair
Burst lowered in exchange for sustained dps through her new W
T.T Leblanc. I imagine (but haven't tested) that she'll be a pain. She is when I play Ahri anyway.

Hello, and welcome to my Katarina Guide. This guide is based on the knowledge I've come to gain from playing Kat mid for a good amount of time, and from the short time I've had the opportunity to come to grips with her new playstyle in a public beta environment. As such, this guide hasn't been finished as I have yet to truly test the new Kat beyond 1v1's, 1v2's, 2v2's, and a single troll game, where I was forced to jungle xP. We won't talk about that game. *shudders.

The point I'm trying to make here, is that Kat is a viable choice, and I've stuck with her for a loooong time. So I'm hoping my guide will be well written enough for you to learn something :)

Aside from the 1 point left over, which I threw into the +1 health regen mastery (for a lack of it being useful anywhere else, as our jungler doesn't typically give me blue, and I've never particularly had a sudden feeling like. Damn. If only I'd had blue buff that extra shunpo, bb, ect. could've done something. So I put it into health regen. I'm building Kat to be a bursty assassin with sustain. My goal is to outsustain the enemy mid or top, push their lane back, and then gank another lane or farm and get ahead. My sustain comes from early spell vamp, and 9% at level one is nothing to shake a stick at! That transitions into 29% when I get a Hextech Gunblade.

Why cooldown reduction? Why not magic resist for enemy midlane? .-. Because as a Katarina, you shouldn't EVER be hit by something, and if you are, just heal it up and push them back. Don't be stingy on potions. Or wards. They both save lives. I get Cooldown reduction because, combined with 4% from masteries, you end up with 10.5% free cdr, which is good for a champion who's cooldown based. CDR isn't as important on Kat though, because her cooldowns are really low (6 seconds on Shunpo at max level and 50 seconds on ult at max level) so you can focus on damage and health items.

I get armor because 9 times out of 10 it's the AD carry on bot that gets fed. I never let mid get fed because I ward, call mias, and kite like no other :3 Videos will be added at a later date that showcase what I mean by this.

I max W first because it provides a movement speed buff, (let's me wait for boots and kite like a maniac) procs my Bouncing Blade's mark, and provides an AOE. This all amounts to easy laning, and a lovely getaway when combined with intelligent uses of shunpo.

After my W, I can max either Q or E first.

Why do I usually max Q first? Well, I agree that having your Shunpo cooldown drop from 14 seconds to 6 is amazing, but using your Q aggressively will both give you sustain in lane (spell vamp) and push your lane, last hit, and harass. Shortly after you start maxing your Q laning phase will begin to end. You can max your E first, it provides you with peace of mind, knowing you can always pull out your get out of jail free card every 6 seconds, but an excellent Kat player will know when and when not to blow her E, and can instead focus on harrass, zoning, last hitting, and ganking effectively.

I start Boots and 3 health pots because mobility and sustain are half of what makes Katarina who she is. I don't use Doran's items because Doran's Blade doesn't affect her Q, the lifesteal is useless early game, and the mana regen from Doran's ring is useless on her.

I get Hextech gunblade for the AD scaling on her W and R, the Spellvamp, and the active slow and damage on it, which adds to her burst :3

I get Rylai's for the 80 AP, the 500 Health, and the slow, which allows me to kite even easier than before. Plus, a rod of ages would've left me with useless mana and 10 minutes until it was ready to be used to its full potential.

I get Rabadon's Deathcap because it's a huge burst to AP :3

I get Sorc Boots because 20 magic pen is never a bad choice when it comes to boots. But if they're heavy on CC or AP some merc treads would be a lovely choice~

After that, build to your liking. And after you reach level 10 or so, don't be afraid to get elixirs, Blue and Red Elixirs work wonders on her ( AD scaling is great on her W and R, and CDR+ AP is amazing :3)

When farming and Harrassing, the goal is to throw your bouncing blade, W the enemy laner at max range while consuming the mark on the minions nearby, and then running back to dodge their counter harrass. It sounds complicated, but it isn't.

Important things to keep in mind:

Don't use Shunpo only as an escape tool. 20% reduced damage is HUGE. When you go in for the kill, don't be afraid to use it.

Don't throw your Q too early, they can easily kite you if you don't proc your W on them. Use it fairly last second, or combine it with your Shunpo + E

When going for the kill, wait out of sight or shunpo onto them. Q them, (Q as early as possible, if possible do it before you shunpo, the point is to have it up again after you ult in case they live) W, Ult, then when they are almost out of range, shunpo onto them, W, and stick with them until you kill them, or you can just let them go. Having them get pushed out of lane lets you either free farm, get their tower, or gank another lane. :3 Don't see it as "a lost kill" but merely as what it is. Them not getting any farm for a while.
Farm with Q + W.
If they're about to jump on you, use your Gunblade's passive, 300 magic damage and a slow (plus a potential exhaust) are very. Very. Painful things.

I run Exhaust and Flash. Why? Because sometimes I need an uber amazing escape ( Shunpo onto wraiths or dragon, then flash over the wall ftw) or I really need to shut someone down. Exhaust not only reduces their movement speed, allowing me to kite them all day long, but it also shuts down their damage output, turning any fight around in my favor. To add to that benefit, it also reduces their Armor and Magic resist. :3 very nice!

Ignite is always a viable option though. Same with Ghost and Teleport.

Sorry that my guide's incomplete, the goal here was to get up the items, masteries, ect. So that newcomers to Katarina or older players of Kat would know how the recent changes have affected her, and how to play and build her now :3
This is just how I build her, and I've left 2 open slots for more items. :) I'll revisit this guide as I get some more hours clocked with the newest reworked champion to be introduced to the League of Legends. Thank you all for reading my guide, and please, stay tuned for more updates. This does lose its viability without spell vamp runes, so YES they are ABSOLUTELY necessary :) I'm sorry. If you can't afford them or don't want to try it, look for another guide. You can't simply play using a build if you skimp out on items before the game has even begun~