In "My Last Duchess," when the speaker concludes his memories and complaints about his late wife, he turns his attentions to his upcoming marriage. He addresses the envoy and says they will rejoin the party downstairs. He notes that his master has a reputation for being liberally generous ("known munificence"). This is proof ("ample warrant") that his claim (pretense) for a dowry will be granted. That is, no claim for a dowry will be disallowed. Then the speaker tells the envoy that it is the daughter herself, not the money from the dowry, that he is after. Judging from the speaker's prior comments about his late wife, the reader (and perhaps the envoy) will have certain doubts about the speaker's good intentions.
Why do you think the Duke shows the envoy the painting of the former wife who was clearly murdered? Is it a threat or a warning, intended to influence the future wife's behavior? Or is the Duke so blinded by egotism that he merely wants to show off his deadly accomplishments? Browing leaves this point ambiguous, and I think it could be read in multiple ways.