In Melody mode there are to ways to add chords to the chord table.  The first is to use the calculator buttons -- Add, Insert Before and Insert After --, and the second is to click on an already selected note with the right mouse button.  In other words, double clicking on a note on the staff with the right mouse button adds the chord to the chord table.

Remember, FastChords calls a "chord" the state of the keyboard, whether the keyboard has actually a chord proper (i.e. in the music sense) or not.

One important way in which the Melody mode differs from the Chords and Table mode is that when a chord is added to the chord table in Melody mode, if a chord exists on the keyboard it is erased.  Only the melody note (if it exists) remains. 

Important Shortcut:  clicking twice with the right mouse button on the same staff note adds the melody note (and any chord notes present on the keyboard) as the last chord in the chord table.  This is the fastest way to enter the melody lines of the song.  See Entering Melody Notes in  The Best Way to Enter Full Songs.