I HAVE TO EDIT THE GUIDE. DO NOT CRITICIZE ME ON THIS!!

Begin the game by buying an Amplifying Tome and a health pot, and getting Shunpo as your first skill. Katarina is a strong laner @ level 2 and a strong solo laner after about level 3 or 4. Regardless of if you solo or not continue the skill sequence as stated.
Depending on how well you farm minions, by level 6 you should have enough gold to get the Hextech Revolver. This is a crucial first item in my build due to the Spellvamp it gives. Early game this will give you a slight advantage over opponents who have little to no healing or magic resist. Late game the spell vamp becomes a big advantage regardless.
Once you hit level 6 (given you're not laning against someone with any type of CC that can interrupt Death Lotus) you want to start playing more aggressively, considering if you use exhaust and rally at the right time you will basically trap your opponent in your ultimate and they will die.
When initiating a fight you want to first take into consideration how you are going to use your Killer Instincts. If your enemy has healing, using it in turn with bouncing blade will ensure they won't heal nearly as much, however if they don't have healing using it with Shunpo would be best, although if you're going to use it with shunpo take into consideration if you thin they will fight or flee. If you think they will flee save killer instincts for the chase, because if you're going to turret dive the reduced damage will likely save you from the tower, however if they are low enough for bouncing blade to kill them, but are out of range, using Killer Instincts with that and targeting a minion could easily land you the kill w/out the risk of execution via turret. Before fighting let the lane get pushed, then when you have an opening shunpo to them. DO NOT immediately throw down rally and exhaust. After shunpo run a few steps behind your enemy to put yourself between them and their tower, then use your spells. This is so that if they want to be 100% safe they have to run through your ult to their tower, or run off into the jungle where they may be ganked by other team members.

Around level 11 you should have enough gold to buy your Abyssal Scepter as well as your Spell Pen. boots. By this level your team should be looking towards initiating team fights. When your team does DO NOT JUMP IN UNTIL AFTER THE FIGHT HAS BEGUN AND ALL THE CC HAS BEEN USED. YOU WANT TO BE THE LAST ONE INTO THE TEAM FIGHTS SO THAT YOU CAN USE YOUR ULT TO ITS FULL POTENTIAL.
When you do jump into team fights you want to use killer instincts with Shunpo, since it reduces your damage taken it will blur out your champion for the duration of the effect. Immediately throw down rally, but save exhaust for someone that tries to run. Once your ult is finished shunpo to the nearest champion and chase carefully.

*NOTE*
You may have noticed that i have more items than you can actually build in a game. This is because i alternate between getting a gunblade, will of the ancients and deathfire grasp depending on the game. All three are viable items for katarina, although due to insufficient time between building and finishing the game i find myself not having time to finish the gunblade and i just get a will of the ancients for the aura.


I hope this guide helps. I have used this build (although did not fully complete it) and have gotten countless Quadra-Kills, 1 Penta-kill, and many many many Triple Kills during 1v3's in 3v3's as a carry.