Charlie Swan and Sue Clearwater entered in a relationship in Breaking Dawn, and are presumably two of the last characters to do so.
In the films, Charlie Swan is portrayed by Billy Burke and Sue Clearwater by Alex Rice.
Prehistory
Charlie Swan
Charlie Swan is the Chief of Police in Forks. He was previously married to Renée Higginbotham, and fathered a daughter named Bella. Sue's husband, Harry was one of his best friends.
He did not remarried since his divorce with Renée. Charlie visited Renée and Bella once every summer when Bella reached adolescence. His daughter married Edward Cullen a vampire, at age 18, and gave birth to a hybrid, Renesmee, before becoming a vampire herself.
Sue Clearwater
Sue Clearwater is a nurse and the mother of Leah and Seth Clearwater. She and her husband, Harry Clearwater, were both of Quileute descent and knew of Seth's eventual fate. However, when Leah also phased, it shocked them completely as she was not expected to. It was so impacting that her husband had a cardiac arrest, and died of it.
Sue learned to remain strong for her family, and quickly overcame her pain.
History
Breaking Dawn
Charlie and Sue become much closer sometime after Bella Swan and Edward Cullen's wedding. By learning about the supernatural world, Charlie begins to adjust to the pros and cons of it, and Sue is constantly by his side for support.
Whenever his newborn granddaughter does something too advanced for her age, he would look to Sue for support. Sue keeps him company and cooks for him.
On Christmas, she and her family spend it at his house, along with Bella, Edward, Jacob and Renesmee. However, as their relationship improves, Charlie's friendship with Billy Black, his best friend, begins to waver, probably because he is dating a dead friend's wife.