After much deliberation and reading her exemplary Champion statistics from lolbase, I declared Katarina the Sinister Blade number one as the most overpowered character for winning solo queue normal games. Here is a link if you want to see the entire list: the top ten pubstomping Champions and how to counter them. She is still easily countered in ranked, but if you feel like facerolling casuals with an OP Champion, try her.
"The Sinister Blade of Noxus is like a black widow - beautiful, but deadly." –- Garen, the Might of Demacia, following a skirmish in the field .
I don't think anyone can dispute the result of a fed Katarina versus a disorganised team, her passive makes her the ultimate snowball in a team fight. However, just because she's overpowered doesn't mean you can hurr durr into the enemy team and get kills, if you want to do that play something like Nunu or Jax.
Here is a video of my Kat play in normal matches:
This footage is by no means a display of skill, it is just some ganks and gameplay.
As a militaristic state, it would perhaps come as no surprise that the primary responsibilities of Noxian women are raising strong children and providing succor for husbands who walk the path of soldiery. There are some, however, for whom nurture is not nature, and Katarina is just such a creature. Born the daughter of the feared Noxian General Du Couteau, the girl was always more interested in her father's knives than in the dresses, jewelry, and other trifles that her sisters spent so much time fussing over. A childhood dispute soon unveiled her uncanny knack for bloodshed, and her father, ever the opportunist, happily fostered her killer instincts. After training under the tutelage of the finest assassins the mighty city-state had to offer, Katarina first cut her teeth performing assassinations in the Ionian War. There, her ruthless mastery of the knife and dagger combined with her sadistic temperament to earn her the title Sinister Blade, a moniker that would follow her for the remainder of her career. She is most famous, however, for her exploits in the campaigns against Demacia, especially those surrounding her engagements with the nation's champion, GarenSquare.png Garen; a rivalry which began when she recovered the remains of Sion from a vanguard under his protection.
With the demand for warfare somewhat arrested by the tenuous peace settling over Valoran, Katarina sought the allure of violence in its last remaining haven: The League of Legends. Some might say it was an inevitability that Katarina's chosen path would draw her to become a champion of the League, continuing to enforce the will of Noxus and bring prestige of war to her noble family. There, she continues to perform her bloody work - esteemed and reviled, feared and adored.