To write a critical appreciation of "My Last Duchess" by Robert Browning you need to compose an essay which goes beyond mere content summary to analyse the literary features of the poem. Your starting point should be the genre of dramatic monologue and Browning's habitual method of using the narrative voice to undermine itself by gradually revealing information or attitudes at odds with the image the narrator is trying to present. The crucial lines, in which the Duke appears to be revealing that he killed the Duchess are:
Oh sir, she smiled, no doubt,
Whene'er I passed her; but who passed without
Much the same smile? This grew; I gave commands;
Then all smiles stopped together.
You might also examine the use of highly enjambed iambic pentameter couplets as synthesizing the elevated manner which indicates the Duke's artistic refinement with a conversational tone.