I would argue that Formalist criticism is the best approach to analyze Robinson's memorable poem.
Biographical criticism would point us toward Robinson's life to look for clues about the poem's meaning, but this leads us away from what the poem actually seems to address. Historical criticism, where we contextualize the poem within its historical moment, seems unnecessary in this case: this poem, with its message, can be understood perfectly well outside of this context, and this accounts for much of its power. There isn't anything about the poem that suggests it would be best approached via gender criticism; similarly, the same argument can be made about psychological criticism or reader response theory.
When we employ a Formalist approach, we study the text and only the text—ignoring its author, its historical context, its other potential readers, and so forth—emphasizing the importance of the poem's form and meaning. The ending of "Richard Cory" is likely a surprise to most first-time readers, and it compels us to return to the beginning of the text and reread it, looking for clues. These clues to why Cory might go home one night and "put a bullet through his head" are few, and we must discern the subtext to understand that, despite his "imperial" appearance, his "grace," and the way he "glittered," Cory must have been incredibly unhappy. Perhaps others' perception of him as "everything" enviable translated to a terrible sense of isolation for Cory himself. We cannot, therefore, judge others by their appearance; we never know what is going on within another person's heart and mind.