Viewers never meet Harry Morgan when he's alive.
This is Dexter's foster father, who adopted the boy after his mother was murdered as part of a failed drug bust. Harry is often seen in Dexter's flashbacks, as we learn the lessons that he tried to impart on his son in order to assist him with his rage and murderous tendencies.
We also learn that Harry was actually having an affair with Dexter's mother - a criminal informant - at the time of her slaying. He killed himself once he saw just how far over the edge Dexter really was when it came to ritual killings.