Who is speaking? To whom are they speaking?

The speaker in this poem gives away very little about themselves—we do not know whether the speaker is a man or a woman, how old they were, or where they were from. However, we do learn one very important thing from the first line, "I heard a fly buzz—when I died—"
The speaker, then, is a dead person or a ghost; the poem is being relayed to us from beyond the grave.
We do not know from the poem who the intended audience might be. However, we do know that the speaker is somebody who was much beloved in life; "The Eyes around—had wrung them dry—" suggests that the speaker's deathbed was surrounded by mourners, all crying over the loss of this person. We also know that these people were the recipients of "Keepsakes—Signed away," so the speaker was presumably not somebody too impoverished to have had possessions enough to leave to the mourners in their will.
Apart from this, the speaker may be any dead person, and the poem is addressed directly to whoever is reading it. Dickinson's conversational style always lends a sense of immediacy to her work, and for this reason it is possible for us as readers to imagine that the dead speaker is relating this story of their deathbed for our own benefit.