Relationship with Victoria
We first see Lord M in Doll 123. He offers his services as an advisor to Victoria, and at first she refuses. Victoria and Lord M's relationship becomes strained after Victoria realizes he is the only man she would ever want to marry. Lord M and Victoria eventually part ways after she marries Prince Albert.
At one point we learn that Lord Melbourne was once married and had a son. After his son tragically died, his wife left him and he never saw her again.
Later life & Death
Later in the series Victoria's friend, William, retires to brocket hall and works on his garden. After hearing that he is ill, Victoria brings him a toy bird that sings a melody. Following her visit, she goes home and finds her dog dead. During the funeral of Dash, we see Melbourne dead staring at the bird and the bird looks beaten up.