I cannot be a man with wishing; therefore I will die a woman with grieving.
I'll devise some honest slanders to slain my cousin with.
Much Ado About Nothing is a lot like high school romance. Everyone knows everyone else's business (or at least they think they do), and everyone is trying to play matchmaker and fix each other up. So, sometimes people get the facts a bit wrong — which can lead to some humorous misunderstandings — but they basically always have good intentions.
The thing to keep in mind about Beatrice and Benedick is that they love to trade verbal zingers with each other. They're constantly arguing and competing to see who can come up with the cleverest come-back line. All of their friends can tell that they're attracted to each other . . . except for Beatrice and Benedick. Maybe you've known people like that?
Let's take a look at those two quotations in the order that they happen in the play — beginning with the second one first.
The quotation occurs during a conversation between Hero (Beatrice's cousin) and her attendant Ursula, in which they're scheming about the best way to play matchmaker between Beatrice and Benedick. Ursula discovers that Hero has heard from some of Benedick's friends that he is in love with Beatrice. Ursula figures that she should just tell Beatrice as much — but Hero doesn't want to do that because she's afraid that Beatrice will just laugh and reject Benedick altogether. (Beatrice just has that kind of prideful personality, I guess.) So, Hero makes the sarcastic comment that what she should actually do is say something negative ("honest slanders") about Benedick, because that would probably be a more effective way of getting Beatrice's attention. "One doth know how much an ill word may imposition liking." Ursula is horrified at the thought, but Hero is determined to come up with a sneaky way to fix up her cousin.
The other quotation occurs later in the play when Beatrice and Benedick are having one of their typical flirtatious arguments. In the course of their conversation, Beatrice refers to Claudio (their mutual friend who is dating Hero) as her "enemy"; so, Benedick calls her out on it and asks her to explain what she means. That really ticks her off because Beatrice feels that Claudio has dishonored her cousin Hero. Beatrice basically tells Benedick that he's being a wuss and that, if he were a real man, he'd stand up for Hero and put Claudio in his place.
Furthermore, Beatrice would really like to do something herself to stick up for Hero, but she feels like she can't because she's a woman and it wouldn't be proper. Beatrice is thoroughly annoyed that the men (specifically Benedick) are too wimpy to take action — so she says that she wouldn't want to "be a man with wishing." But, on the other hand, she can't really do anything worthwhile as a woman, except to just "grieve" and feel miserable — and if she has to do that she feels like she'll literally "die a woman with grieving." It's pretty melodramatic and sarcastic — but you can tell that she's really upset.
There are lots of other things going on in this play, too — but hopefully this will help shed some light on these two puzzling quotations. Whenever a character says something strange or contradictory, ask yourself whether he or she might be speaking sarcastically (it happens a lot in this play). This cast is sort of the Dawson's Creek equivalent of the Shakespeare world.