The Speaker
The conceit of the poem is that the speaker is addressing his lover, though we’re not meant to understand it as an actual spoken or written communication. Other than the fact that he is in love, we know next to nothing about this speaker. This is the conceit upon which the entire poem turns: that the speaker and his betrothed were never really alive until they found each other, and all that came before was just a dream to prepare them for the real deal. Ultimately, it can be said that the speaker and his lover are not two distinct and separate characters, but exist as one, projecting mirror reflections of the other—each only one half of a circle that is only complete when joined together.
The Lover
Though the references to religion, and the importance of Donne's faith to his work in general, raise the possibility that the “love” described in “The Good-Morrow” might be seen as religious in nature, ultimately the poem makes clear that the lover is as mortal as the speaker. Most significantly, the experience of awakening the poem describes is understood to be shared by both the speaker and his lover, which wouldn’t be the case if “thou” were a stand-in for God or Christ. Other than this, however, we’re told next to nothing about the speaker’s lover, though from context it’s safe to assume that his love is reciprocated, rather than one-sided.