Driven by an intense killer instinct, Katarina uses her talents as an assassin for the glory of Noxus, and the continued elevation of her family. While her fervor drives her to ever-greater feats, it can sometimes lead her astray.
From childhood, Katarina displayed a natural gift for combat. As the daughter of a prominent Noxian general, many paths were open to her, but she rejected them all for the path of the blade. Rigorously trained by the finest assassins in Noxus, her father the best among them, it was not long before Katarina demanded her first assignment. The task they gave her was aggravatingly simple: assassinate a low-ranking Demacian officer. As she set to her work infiltrating the enemy camp, Katarina discovered an opportunity too tantalizing to pass by – the arrival of a Demacian General. Stalking him to his tent, she quietly dispatched his guards and slit his throat. Pleased with her impressive kill, she disappeared into the night. Katarina’s elation faded the next day when her original objective, the Demacian officer, led his forces to ambush unprepared Noxian soldiers. Though the Noxians fought valiantly, they suffered heavy casualties. Furious at her mistake, Katarina set off to complete her original task. Returning to the camp, she spied her now heavily guarded target and realized a stealthy kill was no longer possible. Drawing her blades, Katarina swore the officer would die, no matter the cost. She leapt into battle, unleashing a whirlwind of steel. One by one blades flashed and guards fell, each strike bringing her one step closer to the officer. A final thrown dagger restored her honor. Bloody and bruised, Katarina barely escaped the Demacian forces, and returned to Noxus a changed woman. The scar she earned that night now serves as a constant reminder that she must never let passion interfere with duty.

PROS
Manaless champion.
Shunpo allows you to be very mobile.
Great AOE damage.
Snowballs pretty hard.
Fun to play.
CONS
Squishy with this build.
Can be focused hard if the enemy team is smart.
Falls late game (35-45 mins).
Melee range.
Her Death Lotus can be interrupted.

Starting Items
Due to Katarina's weak early game,she needs sustain in lane so that's why we got health pots.The enemy AP carry is gonna harass you all day if you stay too close to last hit with AA.The wards are very helpful for escaping ganks and for seeing where the enemy jungler is.:)
Core Items
This is my standard core build when i am playing Katarina.Personally i build this items depending on how the game is going.If i am doing well in lane ---> Rabadon's Deathcap;however if the enemy laner is dominating me i rush---> Zhonya's Hourglass.If you are having trouble vs an AP mid pick Negatron Cloak. Deathfire Grasp should usually be bought as the last item, since it only brings single target damage and isn't as effective in mid-game.IN RABADON WE TRUST
Endgame Items
Ok.So let me explain the importance of each individual item.
Sorcerer's Shoes Gives you flat Magic Penetration, which, tied with your runes and masteries, will let you deal near-true damage to anyone before they get MRes, which usually isn't until mid-late game.Standard choice.
Rabadon's Deathcap is the "bread and butter" item for almost every AP carry.
Zhonya's Hourglass acts as a bait item, when Kat gets into later game her spell combo will not be able to kill the enemy, having an Hourglass allows you to not only wait for your cooldowns to get off CD, but to bait the enemy into blowing skills toward your direction.
Guardian Angel is like Zhonya's Hourglass v1.0.Nothing special to say about this item.
Deathfire Grasp helps you to burst down carries.The reason why i dont rush this item is because it's risky(you are extremely squishy and it's harder to make "the plays").

Void Staff is used to counter the enemy's magic resist.This item is necessary to pick up late game. More often than not, when the enemy sees your crazy damage, they will be buying some Magic Resistance to counter you. How to counter them countering you? Void Staff

ABILITIES
Voracity
Champion kills or assists reduce Katarina's cooldowns by 15 seconds.
This is what makes Katarina fun to play. Generally your role in teamfights is to jump on low hp enemies.
Bouncing Blades
Throws a dagger dealing 60/85/110/135/160 (+0.45*AP) magic damage. The dagger bounces to the 4 closest enemies dealing 10% less damage with each bounce.

Enemies hit are marked for 4 seconds. Katarina's basic attacks or spells will consume the mark dealing 15/30/45/60/75 (+0.15*AP) additional magic damage.
Range
675
Cooldown
10/9.5/9/8.5/8 Seconds
This is the ability you want to max on Katarina.Maxing Q is best for harassment in lane,farming from a decent range,poking in mid-late game pre-teamfights.
Sinister SteelSinister Steel
Whirls daggers in a circle dealing 40/75/110/145/180 (+0.6*AD) (+0.25*AP) magic damage. If she hits an enemy Champion, Katarina gains 15/20/25/30/35% movement speed for 1 second.
Range
400
Cooldown
4/4/4/4/4 Seconds
Low CD AoE damage spell.Helps Katarina waveclear during the laning phase.The movement speed buff allows you to catch people and also to escape.