Well, since the remake, I've been hearing a lot that Akali is an impossibly hard counter to Katarina. (I'm an ex Kat main who switched to Akali b4 quitting entirely).
Here's what you do:
NEVER push the lane
lvl's 1-5: dodge her Q, lasthit with your Q. Don't let her Q you.
IF she Q's a minion (which won't bounce on you), deny her from lasthitting anything. Basically you pretty much spam your Q. Either lasthit a minion or harass her.
lvl 6: harass her every time you have 2 stacks of your ult. She can't possibly trade with her, and you can ult out of her ult instantly if she uses it.
Old Kat shat on Akali without troule. New Kat can't ever trade with Akali. The only thing you have to watch out for is to not fight Kat when she has her ult, and you have less than 2 stacks of yours. Even if she deals slightly more damage than you, you have sustain and she doesn't.