This is not a strict analysis but just some dotted thoughts. Feel free to comment your own thoughts or disagreements on any points I make.
Lady Lazarus feels like it belongs with Daddy. I don't mean that they cover the same subject, but that they feel like chapters in the same book. While Daddy illuminates facts about the narrator's life through the lens of talking about another person (her father), Lady Lazarus takes a much more direct approach.
It's mostly about suicide.
At the beginning:
"I have done it again.
One year in every ten
I manage it--"
What has she done again? A suicide attempt.
And then we have a few stanzas which show how... uneasy she is about herself? About life?
Lines such as: "A sort of walking miracle, my skin / Bright as a Nazi lampshade" Are certainly not positive feeling images. Sour breath, peeling skin, it all has a quality of death and uneasiness. Then she says "soon, soon the flesh / the grave cave at will be / at home on me", basically saying she's close to attempting suicide again. The "peanut crunching crowd" feels like it's Mr Gordon, "unwrapping" the narrator.
There's some dark humor halfway through: "Dying / is an art, like everything else / I do it exceptionally well". Of course we know she doesn't do it exceptionally well.
Crudely, in the second half of the poem the narrator talks about how she dreams(?) of committing suicide: burning herself alive ("Melts to a shriek / I turn and burn") and letting others find nothing left of her but ash and the jewelry she was wearing.
The last stanza also seems very important:
"Out of the ash
I rise with my red hair
And I eat men like air"
This sort of resembles the discomfort Esther has for men in a more extreme way. It feels like the whole red hair rising out of ash could be a allusion, or maybe she just used red hair because it relates to fire.