That Godfather scene is just prior to him calling the undertaker/mortician and asking him to "clean up Sonny".
The undertaker is actually first seen in the movie in the opening during the wedding as the man asking for a revenge killing for his daughters honour, to which the godfather replys that no, no one will be killed for this but the men involved will be beaten for dishonouring the mortician's daughter. The godfather even chastisizes the mortician because has never really acknowledged the godfather nor involved himself in any business until he needed a favour. The mortician understands he now owes the godfather a debt.
When the mortician is first called for repayment to the godfather he is not told why and worries, as he readies himself, that his repayment will involve him in a major crime. He then finds out it is because of Sonny's death that the godfather has called. Upon hearing this news he tells the godafather that he cannot accept this job as repayment for his debt because he accepts it as man who is also a father.
The movie never really plays out quite a few entire stories unless you've gotten an out-take disc and the book.