One of the major differences between a post-graduate and an undergraduate level essay is that in graduate school your work needs to reflect knowledge of the major critical approaches to your subject. Your first step, therefore, after reading the poem carefully, is to search the MLA International Bibliography (available on your school's library website) for recent scholarship about Plath.
Because of the intensely personal nature of Lady Lazarus, and Plath's own psychological history (as documented in her novel "the Bell Jar", Plath is often the subject of psychological (both Freudian and Lacanian) or feminist/gender criticism. The strong evocations of the Nazi also opens fruitful possibilities for linking Plath to the burgeoning field of Holocaust Studies, especially from the point of view of reading facism and its erasure of the other as congruent with patriarchy's erasure of the feminine (Kristeva might be useful here).
Another area to explore is the relationship of Lazarus, a character in the New Testament who is raised from the dead, with both Nazi oppressors and Jewish victims. What is the function of Christian iconography in the poem?