You bait skillshot champs into taking shots at you - then dodge with Shunpo at a minion (Yours or theirs) as worst-case scenario, or you shunpo direclty onto them, or you dodge on foot and get into shunpo range.
Then you hit your Q and W as a combo, maybe land an autoattack, and bolt away.
You don't trade. You dodge and punish. You're a counterpuncher.
People got used to betting on lucky Qs. Don't do that anymore. Get used to predicting ricochets. Play some Miss Fortune to get used to the idea of it, then get used to how Katarina's ricochets work as opposed to those (I already main MF as my second-best AD carry, so, I'm used to abusing Double Up. Especially since I played Release MF: Where it would be MF mid against Ashe mid. AP carries almost -never- went mid back then).
It's a different Kat, and the luck element is gone. A lot of people relied on the luck element of BB and got used to it. You need to be better to get the same performance out of new Kat as old Kat. But she's better overall.