This is how to play Katarina, period.
I play Kat a lot. She is my main.
I have Team On My Back to back me. LOL
Plus I have my gameplay on XorexFilms to agree.
Link to XorexFilms is HERE


Any questions?
PENTAKILL CAN BE FOUND AT THE BOTTOM

Go Mid, and if not, then go Bot (preferably with a support).
Build 1 is for when a or a few champion(s) have built magic resist.
Build 2 is for when the enemy team has little to no magic resist.
(If you are getting CC'ed, buy a temporary Moonflair Spellblade.)

In Dominion go Bot and recall back after you capture point for boots.
Build 1 on Team 2 is for all dominion.
Build 2 on Team 2 is for emergencies, ONLY if the nexus turrets are down for either team.

Pros
Burst Damage is Insane
Spell Vamp gives health back
AP carry little cooldowns
Multi-target ultimate
Slow from Rylai's
Great Chaser
Great with Blue (More cooldowns)

Cons
You will get CC'ed, but that's what Zhonya's is for.
Can be considered squishy
Easily stopped ult
Can't move during ult
Targeted to death

For the noobs. We love you.

AP - Ability Power
AD - Attack Damage
CC - Crowd Control (stuns, slows, fears, etc)
MR - Magic Resistance
HP - Health Points
CDR - Cooldown Reduction

What You Should Get

Flash
Even after the nerf, flash is still the most viable and multi use summoner spell in the game and with the mastery, is able to be used in tons of different types of situations, such as chasing or escaping. The idea that Kat already has a flash with her shunpo is a misconception due to her not being able to shinpo unless it is to something.

Ignite
This is the second best summoner spell for any mid summoner. Ignite is a super plus due to the new mastery for it, which grants passive AP when not on cooldown. Ignite is also great due to the tower dive ability of Kat that everyone loves. Especially when an enemy champion is a squishy carry and on low health and running (Unless they have a Soraka or Shen half way across the map).

What You Can Get

Cleanse (Anti-CC as well as now Ignite)
Exhaust (Great for Chase and Escape)
Heal (Only way to help team besides kills)
Ghost (Great for Chase, Escape, and Rushing)
Teleport (Teleport Ganks are OP, ROFL)

What Not To Get

Garrison (Awesome turret diving disregards this)
Revive (You shouldn't die you're KATARINA)
Smite (She is not a jungle, well not really)
Surge (Not viable for little AP, and no late game)
Promote (She does enough damage already without it)
Clairvoyance (Leave it to the supports)

Clarity (Well, I guess you could troll and look cool)

Boots of Speed Plus Health Pots

Best for building into better items and allows dodging of spells from mid AP carries.

Hextech Revolver

Gives AP plus early game spell vamp (AP version of life steal) for survivability and remaining in lane.
Just trust me on this one. The extra spell vamp is insane on Kat. It allows her to regain all of her health when she uses her bouncing blades, shunpo, and even her ult. Although Hextech Gunblade may increase her AD and life steal, both only good in her ult, the extra AP and spell vamp is worth it. If you manage to use your ult next to a melee DPS, you spell vamp your health back as he attacks you and loses his.

Ionian Boots

Gives early game cooldown for damage and avoiding ganks, as well as early speed.
Better to have 3 shunpos then 2 shunpos with a little extra damage.

Rylai's Crystal Scepter

Gives AP, HEALTH, and A SLOW FOR ALL YOUR MOVES. Make sure to take advantage of your slow by using your Q, E, and R to keep a constant slow up on your enemies.

Hextech Gunblade

Gives awesome spell vamp for diving into fights and past turrets and more AP.
Also gives AD which scales on bouncing blades and ultimate, plus slowing active with range.

Rabadon Deathcap

The ultimate of AP carry items.

Void Staff

Allows to pummel through magic resist as well as some more AP.

Zhonya's Hourglass

Pop in, ulti, Zhonya's, pop out, game over, you win.
Plus more AP.

Consumables

Oracles allows you not to get ganked.
Elixir of Brilliance gives AP.
Elixir of Fortitude gives health.

Let me know if you get these in games. LOL

Runes = VERY EARLY SURVIVABILITY AND DECENT DAMAGE
Masteries = A LOT OF DAMAGE AND SOME MOVEMENT SPEED

I would suggest, during ranked gameplay, especially when the enemy jungler is a good ganker, to buy two wards and set them up in the brush on each river side of mid lane. Also buying health pots, especially if you get Ionian Boots before Hextech Revolver. Plus let your tank intiate in team fights or wait for someone jump to either who is overextending or has been hit by a decent CC, such as a stun, snare, or knocked airborne.

Obviously, Kat, especially after level 6 has great farming ability with her low cooldown shunpo and bouncing blades. Make sure to use these to last hit minions.

XorexFilms' 100th Video LoL AP Kat w/ Guide + Penta Suicide Play Video