In an older version of Unicode, it was assumed that the Coptic alphabet was made up of the same range of characters as the Greek, with the addition of the extra letters shei, fei, khei, hori, gangia, shima, and dei (03E2-03EF). It has now been recognised that Coptic is a separate alphabet, and the characters are all encoded in a separate Unicode block (2C80-). The Demotic-derived letters are still in the Greek block, however.
As of 2013, the most common fonts used by scholars are Antinoou developed by Michael Everson in cooperation with the International Association of Coptic Studies, and IFAO Copte developed by the Institut français d’archéologie orientale, Cairo.
A set of pages by Moheb Mekhaiel explains how to use Unicode for Coptic (including theory, fonts, drivers, and software advice).
The Pennsylvania State University Teaching and Learning with Technology site has a page devoted to a Coptic Unicode Character chart.