Purple is the color of royalty. In ancient times, the cost of making purple dye was so great that only royalty could afford to buy it. It is said that in order to make just one ounce of purple dye took 20,000 purpura snails, from which the dye was obtained. Purple is also traditionally the color of death, perhaps because it is so dark. Liturgical churches, like the Roman Catholic Church, use the color purple in vestments and hangings for Easter, in anticipation of the Crucifixion.
In her poem, Dickinson uses both meanings of purple. By "mitred Afternoon," she is referring to the fact that most funerals take place in a church (the mitre is a hat worn by a bishop). As she describes the trappings of a funeral: a coach (hearse), footmen (pall bearers), chamber (coffin), bells, attendants, she likens a funeral to a royal procession. In death, all persons are equal, both the highborn and the commoner.