Every recent takedown (kill or assist) she gets resets her cooldowns by 10(?) 15 seconds. When she picks up a dagger, her Shunpo (blink + auto reset) gets its cooldown reduced by 10 seconds or so. Her ultimate is relatively low damage compared to the rest of her kit, and is mainly used to finish people off to enable the rest of her abilities.
The key to fighting Katarina in lane phase is paying attention to if she picks up her daggers after blinking to you, because if not, she's vulnerable.
It's also good to mention that her Shunpo can blink her to friendly or enemy minions, friendly or enemy champions, plants (blast cone, honeyfruit, scrying thing).
She's a very slippery champion, but at least she has relatively nonexistent lane priority as long as you know how to face her.