In this famous poem Dickinson presents her view of death as something that is not to be feared, but as something that confers honour upon us, and also something that levels all humans, no matter what their social class or station in life beforehand. Note how the first stanza introduces the inevitability of death, but also how it confers upon death--normally something that is feared and dreaded--a kind of nobility:
One dignity delays for all—
One mitred Afternoon—
None can avoid this purple—
None evade this Crown!
The dignity of the title obviously refers to death, and the way that we are told nobody can avoid the "purple" or the "Crown" link death to something like a coronation, where "purple" is an unmistakable royal colour and the "Crown" is a symbol of royalty. Obviously Dickinson is challenging popular perceptions of death. From her religious perspective, death was the ultimate "dignity" that greets all humans, whatever their station in life, confering upon them equal honour and glory.
Building on this first stanza, the second stanza contains the reference to "Chamber":
Coach, it insures, and footmen—
Chamber, and state, and throng—
Bells, also, in the village
As we ride grand along!
Dickinson here deliberately builds upon her description of death as being the ultimate accolade by describing the trappings that we receive as we achieve this moment of glory and of honour. In our deaths, Dickinson argues, we gain all of the same recognition and praise that we normally associate with the death of a high-born person, including a coach and footmen and the presence of important people from "Chamber, and state, and throng." Thus through her reference to "Crown" and "Chamber" Dickinson presents death as an equalising force, where equal honour is confered upon everbody.
It depends on what poem and author you refer to. Whitman has a passing interest in the funeral procession but Dickinson writer a poem in her posthumously published work (something about "dignity," I can't remember the title). If memory serves, you're talking about the Dickinson poem.
Dickinson was very religious, so her poetry is shaped by her feelings on religion...one of which would mean a unity with Christ in death (hence the crown and images of "purple"--royalty). For her, the idea of death is that you release the bonds of your mortal life and are on equal footing with all others...no social distinctions, no gender distinctions, you are all united in death.
The 'chamber' statement is a little harder without some additional context and/or the name of the poem so I can go back and read it. If you need further clarification, please LMK the name of the work you're reviewing.