We recommend Sophokeys: http://www.benjaminblonder.org/sophokeys/
After installation a flag will appear in the top bar of your desktop near the date and time. This flag indicates what language your keyboard is currently set to. If you’re typing in American English, the US flag will appear. When you wish to switch languages, you can manually select by clicking on that flag. You can also set a keyboard command to switch languages. Keyboard commands can be set by going to System Preferences —> Keyboard —> Shortcuts —> Input Sources and setting the shortcut for select the previous input source.
Sophokeys covers almost all the necessary characters for editing the Iliad and scholia, and is for the most part a fairly intuitive keyboard to use. Some less intuitive features include capitalization, which is done by first using *, then the character you wish to capitalize.
To type Greek with the built-in polytonic keyboard, see these instructions: http://www.dramata.com/Ancient%20polytonic%20Greek%20in%20Windows.pdf
Another option some project members prefer is EZAccent: http://www.biblicalgreek.org/links/fonts/keyboard.php#other%20options See the instruction manual here: http://www.biblicalgreek.org/links/fonts/Greek_Polytonic_Easy_Accent_Instructions.pdf.
To shift back and forth between Greek and your standard keyboard, type Alt + Shift.