Katarina is my favourite champion you need to be fast with her and u need to remember that her passive resets cd.
On low level when someone is under turret don t be scared to shunpo on him if u kill him just shunpo back on your mob.
Farming is really easy with her just throw bouncing blades and use sinister steel and all mobs are down .
She is great in team fights just don t go first go a bit later because all focus Katarina because her ulti is killing everything in team fights don t spare your cd if u assist or kill someone u ll get refresh of them.
You can take exhaust or ignite.
Anyway good luck with Katarina and have fun.