Kat has a weak lane phase compared to most other midlaners you will face. You need to learn to be able to CS well under pressure, and be able to utilize ward jump defensively to avoid ganks. Your main harass in lane is Q spam, W if you can reach the target to pop your mark and proc the speed boost to back out. You generally don't want to use E to harass because it is your escape, and burning it will leave you vulnerable to ganks, and retaliation from the enemy.
Her teamfighting is strong, especially in lower elo, where people just don't react very quickly to focus cc on you. Kat's strength is in her aoe damage, resets, and cleanup potential. You can rack up a bunch of kill fast if everything lines up for you, but if you and your team lack the damage to get people low for you to reset off of, then you're not gonna have a good time.
One key skill all Katarina players must master, is the ability to properly calculate their damage output at any given time, to be able to assassinate your target with 1 spell rotation, and be able to jump out or go for more kills with the resets you get. A new Katarina player will often miscalculate their damage, and be left with no kills, no cooldowns, smack dab in the middle of the enemy team.
Now I don't play Kat much any more, but I watch Kat streams so I know what I'm talking about. Kappa.