When someone died, their female relatives were responsible for washing, anointing, and clothing the body before laying it out in the house. This was the prothesis, when friends and family could come to mourn and pay their respects. Since the family, any water in the house, and anyone who came in was polluted by the death, a jar of fresh water, preferably seawater, was placed outside the door for people to wash in when they left.** Female relatives sang songs of mourning around the body. During the Archaic Period, all of this happened in the courtyard and often became a way for a family to show of their wealth. All kinds of people came to pay their respects. By the Classical Period, Athenian law restricted these displays, moving them inside to limiting the display of wealth and the presence of women. The only women allowed to be present were close relatives or those over sixty.
Next came the ekphora, or bearing of the body to its grave outside the city. In the Archaic Period, the body would often be accompanied by large numbers of people, often including unrelated women who acted as paid mourners. The women were expected to mourn loudly and dramatically, while the men restrained themselves. Legislation attributed to Solon curbed this display in the 6th century BCE, curbed this display, forbidding women who were more distantly related than the children of cousins and under the age of sixty from participating. The burial itself usually happened just before dawn.
A period of mourning followed. Small offerings might be brought to the grave. The house had to be cleansed and certain family members, mostly the women, remained polluted by the death for a time before they could be purified. The dead should be remembered, but especially by the Classical Period, the time for elaborate or very obvious mourning was over.
*And from what I’ve read, there is a fair amount of continuity through to Modern Greek funeral and mourning practices. Unfortunately, I haven’t read enough on that yet to feel comfortable saying anything about it.
**This contamination was not physical, but was based on the idea that, like birth, the act of dying should be hidden from the gods and anyone who came in contact with it needed to be purified before approaching them.
Lefkowitz, Mary R. and Maureen B. Fant, trans. Women’s Life in Greece and Rome: a Source Book in Translation. Baltimore, MD: Johns Hopkins University Press, 1992. [Especially no. 77]
Pomeroy, Sarah B. Goddesses, Whores, Wives, and Slaves. New York: Schocken, 1995.
Burial Customs, the Afterlife and the Pollution of Death in Ancient Greece - African Journals Online
Death, Burial, and the Afterlife in Ancient Greece - Met Museum