Most folks begin their requests to God with a formal address like "Dear Lord," but our speaker just jumps right in with his questions. He seems pretty anxious about getting to the point and getting some answers.
So who is this guy, and why is he so obsessed with forgiveness? Well, it's never safe to assume that the speaker and the poet are one and the same. With that said, though, Donne himself was a man interested in all matters spiritual, and the tone he adopts in his work is sometimes the same tone we see in "A Hymn to God the Father." He's direct, clever, and maybe even a bit insistent; he repeats "Wilt thou" four times, and then goes on to tell God that, when He's done, the speaker has a few more sins for Him to forgive. Demanding much? (Check out "Batter My Heart: Holy Sonnet 14" to see Donne adopt a similar tone.)
But this consistency doesn't necessarily mean that Donne himself is the speaker, and despite the wordplay with "done" and "Donne," we can't be sure who exactly is meant to be questioning God. In fact, the sins are never made very specific or explicit, so they could represent the sins and doubts of humankind in general. Aren't we all interested in what happens after death?
Our speaker is, that's for sure. In a more general sense, then, it's probably best to think of him as a deeply devout person who turns to God for comfort when contemplating his own fallibility. He seems to be fixated on all the possible avenues for sin that are open to him and is looking to earn forgiveness for any occasion. We think that makes him a bit of a worrywart, really. After all, how thorough can you get when cataloging your possible sins?
Ultimately, he strikes a bargain with God: keep Jesus around and I'll quit worrying. It's the kind of demand only someone who feels a close, personal connection to God can make. He is asking God for a favor, after all. It's not like God's got other things to worry about.