I'm not going to get into how Simone was kicked off the title so this is just thoughts on the preview.
SPOILERISH bit ahead
I'm not sure what to think of Barbara deciding to kill the Joker. Apparently this isn't the first time she's given into the impulse of trying to kill someone? I heard she attempted to kill Mary before she found out the Talon can't be killed. On the one hand Barbara was into some hardcore stuff before the relaunch. I mean she did make her first appearance as Oracle in Sucide Squad after all. I can't recall her ever wanting to kill the Joker personally. She never wanted to help save him and wouldn't mind him dying but going after him herself? It's hard to see. I suppose it's because she gives such a hard time to anti-heroes. Babs could act petty. Since the relaunch she's looked down on most of the Birds of Prey, on them being killers and such. She was also pretty cold to Jason in RHATO. I mean I consider those appearances in character but it's still a little stunning to think she looks down on others for actions she not thinks about but actually attempted.