For me, Katarina is the best champion in the league. Katarina is a melee assassin champion mostly AP (she can either be AP/AD/Hybrid, but I recommend AP) and she has a really high burst damage and AOE damage with all her skills except her Shunpo.
Her main harass skill is Bouncing Blades, but this skill doesn't scale with AD so does her Shunpo, so here's one reason to use AP Katarina

When you are playing as Katarina you must know what skills the enemies champions especially their CC abilities (Croud Control: slows, stuns, fears, snares, roots, etc.) because Katarina is vulnerable to every single CC except slows because all other kinds of CC can literally cancel your ultimate ability Death Lotus, so in Teamfights you MUST wait for the enemy team to use their CC so you can initiate and I must tell you, when a Katarina gets her entrance successfully in a teamfight, she can make difference, but a HUGE difference, BIGGER than a fully stacked Cho'Gath with a Zac mounted on him (It isn't what you are thinking)

I'll be updating this guide as soon as I can. Sorry for the poor information this guide has... for now.