This poem is a dramatic monologue, following both the dramatic and verse definitions as taxonomized by Aristotle's Poetics. It is dramatic in that it has no narrator, and verse in that the speaker is the only "narrator." Approaches to criticism, then, can be dramatic (discussing the ways in which Browning builds a character, showing the complexities of utterances that gradually reveal the true psychological "character" or "noncharacter"of the Duke), or "poetic," discussing the meter, rhyme scheme, connotations, vowel/consonant sounds, references to history or to the development of poetic forms, etc.
Critical appreciation, a vague enough term, then consists of looking at the details of the poem carefully. For example, the last line -- "Which Claus of Innsbruck cast in bronze for me." -- besides following the poetic rhythm and the established rhyme scheme, also is a metaphorical picture of the Duke's attitudes, not only toward his pride in possession (the Last Duchess being among his "possessions") but in his exalted self-view bordering on deification (Neptune taming a sea-horse). The center of the poem can arguably be the ironic line "Even then would be some stooping..." where the Duke displays his jealousy at other people's attention to the Duchess, and resents that she was not wonderstruck by his "gift" of a nine hundred year old name. Then the casual mention of a dowry completes the dramatic imagery and completes the Duke's "character." So by approaching the poem from both directions -- dramatic and poetic ("verse") -- you can intelligently discuss this complex poem.
Thanks a lot for ur guidance