All Francis believes in is work, work and more work which obviously leads to conflict with Peter. It is revealed after Francis' death that he was Peter's stepfather, not his biological father. In the episode "Stu & Stewie's Excellent Adventure" he can be seen attending Chris' Future Wedding, even though he is dead. Due to canon in the series, it is possible that it is another man who resembles him. Also, he would logically not be alive in the future unless when Stewie stopped his near death experience, Francis Griffin's future was somehow altered (along with the futures of other characters) and he would not have originally died. His ghost reappears in "Family Goy".
Although Francis loves Peter, he does hate Lois because she's Protestant, which is why on the wedding day, he taped a sign saying "To A Protestant Whore" underneath the "Just Married" sign on back of the limo. He also hit Brian with his bible for questioning and mocking him. It was after his death that both Lois and Brian celebrated on the yard together that Francis was dead behind Peter's back.
Though Francis is a Catholic, in the episode "The Father, the Son, and the Holy Fonz" he says that Peter's new religion was stupid, but at the end of the episode he is seen praying and kneeling for Arthur Fonzarelli, the character who Peter's religion is based on.