What does the duke reveal about himself to the reader as he speaks in "My Last Duchess" by Robert Browning?

He reveals he had her (his duchess) executed. There’s no doubt that the Duke is self-important. After all, what makes him angry about the last Duchess's behavior is that she thinks anyone could be important as important as he is.

There’s no doubt that the Duke is self-important. "After all, what makes him angry about the last Duchess's behavior is that she thinks anyone could be important as important as he is." that shows how he thinks of himself.

Sir, ’twas not

Her husband’s presence only, called that spot

Of joy into the Duchess’ cheek (13-15)

"perhaps Frà Pandolf chanced to say, "Her mantle laps

Over my lady’s wrist too much," or "Paint

Must never hope to reproduce the faint

Half-flush that dies along her throat" (15-19)

"as if she ranked My gift of a nine-hundred-years-old name

With anybody’s gift. Who’d stoop to blame

This sort of trifling? (32-35)

Those quotes show his jealousy and madness.