Why is Katarina considered an assasin?
I get her design is very Assassiny, but if you look at her actual kit; Her Q is an AoE bounce ability, that sets up her W, which is a circle that deals aoe damage when touched. Her E, her ONLY single target ability, is only even a real damage tool when she's building AD, which isn't often. Her Ultimate is a giant AoE burst damage over time. If you look at her kit, she really isn't an assassin.
Edit;
After rereading this; http://na.leagueoflegends.com/en/news/game-updates/gameplay/dev-blog-classes-subclasses
DevBlog entry by Riot. She seems to be less an Assassin, and more of an Assassin/Battle Mage hybrid.
Assassins are highly single target focused, melee, bursters who have high mobility, and must get into the thick of a fight to kill their targets.
Battle Mages, are near melee range, with the ability to wards off death for a few seconds, while dealing MASSIVE aoe damage.
Katarina, is a melee range burster with high mobility, but is able to skirt around fights and kite defensively while dealing massive AoE damage to the enemy team.