This is an Ability Power (AP) guide for Katarina, her Attack Damage (AD) ratios are better, but her AP effects the main three of her abilities, she is a solo top!.

Flash: I chose flash because of two reasons: 1.) To flash over walls to: Initiate/fight battle, escape turrets, and/or escape chamption. 2.) Flash straight to escape attack.

Ignite: I chose ignite because of two reasons: 1.) To nerf enemy chamption healing. 2.) To kill off enemy champion if they are running with low health.

Cooldown Reduction Gylphs to reduce your cooldowns for better ability "spam"

Health Seals because Katarina has a low health pool early game.

Ability Power Marks so you're abilities do more damage.

Armor Penetration Quintessences so the armor of enemy champs don't get in the way of beating the **** out of them.

I went for "Veteran's Scars" in defense for the health, picking appropriate masteries on the way. In offense, we went for as much damage as possible for your abilities, so AP offense masteries with some magic penetration as well as some +damage dealt seemed fitting, all building up to executioner.

For Katarina, Shunpo is a very key skill, so getting it first is important, after getting Shunpo getting Bouncing Knives wouldn't be a bad idea so you can hit multiple champions/minions in less time, getting Shunpo a second time before getting Killer Instincts would be benefitial to you. After the second Shunpo get Killer Instincts to boost your abilities. Maxing Shunpo first will put you in an intimidating spot, you can do a lot of damage in a very short amount of time. Next, max Bouncing Knives so the knives will bounce on more enemies, and with a lot of AP, that will do a lot of damage. Next, being the utility, you max Killer Instincts last, upgrades don't change the skill much at all. You always get Death Lotus when you have the opportunity.

You start off with Boots of Speed and 3 potions, how a lot of my builds start, so you can be in the lane sooner and when you back/die it won't take as long to get back into the lane, and you can stay in the lane longer the first time around.
On your first back, try to get Hextech Revolver for AP, if you can't get as much of it as you can. After getting that, upgrade your Boots of Speed to Ionian Boots of Lucidity for cooldown reduction. Then get Rylai's Crystal Scepter, then Rabadon's Deathcap, both for more AP. Next, upgrade your Hextech Revolver to Hextech Gunblade to get lifesteal and some AD, if you don't have barely any money when you back, make sure you get Vampiric Scepter if you have to choose between that and a Pickaxe. Then you get Zhonya's Hourglass for AP and Armor, as well as the active it has that makes you invulnerable to any attacks for two seconds, then, finally, you get the Void Staff for AP and additonal Magic Penetration.

Poking early game and avoiding champions is a good idea, once you get Rylai's this rule is disregarded because you have more health, mid game you dominate, jump on overextended champions with Shunpo and spam your abilities until they're slain.

I hope this guide helps you a lot.