This is my Katarina build. I play Kat a TON and I've tested out a lot of builds but this has worked for me the best.

Pros

AP carry with a great early game.
AoE ultimate.
Amazing Cooldown Passive and no mana leads to spell spamming.
Ridiculous damage output.
Shunpo escape mechanism.

Cons

VERY easily countered.
Not incredibly beefy.
Silence is your WORST enemy.
Targetted by skilled players in EVERY team fight.

For Runes I go Magic Penetration Marks and Quintessences. Magic Pen on Kat I feel like is more important early game than AP. Although I rush a Needlessly Large Rod, the Magic Pen runes coalesce very well with the great amount of AP from our first trip back.

For Seals, I run Health Per Level because Kat is a 4 health bar champ w/o items and increasing her beefiness helps intimidate your enemies.

For Glyphs, I go AP Per Level because 27 AP @ level 18 is too strong to pass up. We don't need CDR on Kat because her passive is just amazing.

Your Masteries will be your standard 9-0-21 for a manaless caster (I run the same on Vlad except I run ghost instead of flash) 9 points in offense towards magic penetration and 21 in utility leading up to Presence of the Master.

To start off your game, Katarina is a 4 Health Bar Champion and when you get a Dorans Shield, you have increased armor, Health Regen, AND you will be a 6 Bar Champ.
This is an important first item to get. Your damage output will be insane with this purchase considering you already have 14 Magic Pen.
Sorcerers Shoes on Kat. There really isn't anything better.
Runes + Shoes + Deathcap = Domination. Your damage output is crazy high now, and it should be transitioning into mid game at this time. A deathcap at midgame will make your opponents rage out of their mind.
Slowing your enemies upon success of your Shunpo and proceeding into Death Lotus is an absolute rape combo. This item I am tempted to grab first but the Deathcaps AP passive is too good to pass up first.
The last of your core items. The combo you have prepared is too much to be honest. Your massive amount of AP + Slow + Death Lotus + now -20 MRS? Enemies won't know what hit them.

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A great item on Kat, however my core items are too much of a ***** to stop. The slow on this is great but you want to get directly into your death lotus. If you can slow them with this quick enough, then god bless you, but I enjoy having to just rely on Shunpo + Death Lotus. I don't want to cast the 3rd spell. A great 5th item on Kat imo and who doesn't love the spell vamp?
Oh what a cliche item. This will help your champion be unfocused and will cause 2 spells for you to be un-ulted.

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Your situational item:
If you're being focused during your ult, slip into your hourglass, take the attention away from you, then proceed to dominate with your normal spells. You've most likely assisted your team a little bit so your cooldowns should be refreshed.
The most underrated item in the game hands down. The passive is amazing on Kat especially late game. However the item doesn't give much viability. My core items are all benficial in 3 ways: Deathcap gives AP, AP passive on itself, and AP passive on your other items; Rylai's gives AP, Health and the slow; and Scepter gives AP, MRS, and the -20 passive. The only reason I don't assure this item in my build is because of that.
A great team donation item. Plus having close to 50% spell vamp with Death Lotus is ridiculous.

Max out Bouncing Blade first as it will give you amazing success with your farm.
Max Shunpo second because the damage output is overall greater than Bouncing Blade, however your farm is much more important.
Max Killer Instincts last because it is not as important as your damage inducing spells.
Max out Death Lotus as you would any other ultimate. 6-11-16

Take mid. Do it. Kat @ lvl 6 is the most helpful in ganks/teamfights.

Don't worry about lacking creep kills as a melee. Your farm will be easy with Bouncing Blade.

Harass frequently with Shunpo and Bouncing Blade.
Your Harass combo will be E+W+Q+run behind your creep wave. They'll target you once you've shunpo'ed to them but for the most part, your damage output will beat theirs. Plus you have your Doran's to regenerate what you've lost.

Your 1v1 combo will differ but not by much. If you're going against a healer, you will want to go E+W+Q+R. For the 50% healing by Bouncing Blade.

If you're going against a standard champ, you'll go E+R+W+Q, using Bouncing Blade to clean up what mess you've made already.

In team fights,you NEVER want to initiate team fights. You want to lurk behind your team and initiate 2nd or 3rd.
Let's say your team is:
Katarina
Caitlyn
Xin Zhao
Malphite
Soraka

Malphite will initiate and xin will jump in. Your turn comes now. The focus will most likely be on xin and this is your time to shine. Never initiate or your team will rage.