Major William Morgan is an agent with the American Military Intelligence.
The rank shown is accurate and current; he routinely dresses in his uniform when at headquarters. In the field, understandably, he wears whatever civilian clothes are appropriate. It is obvious that Morgan is a career man, being roughly 40 years old when we first meet him and clearly comfortable in his position. The confidence that he routinely shows on his assignments as well as the respectful ways others treat him tells all there needs to be to shown; Morgan likes what he does.
The exact timeframe of the activities we are allowed to follow is not known. Certainly if the missions took place near the time we are allowed to watch them, 1955, this type of clandestine work would be done by the CIA, not Military Intelligence. However, these assignments could have easily taken place in the first few years after the end of World War II. From a few of the nearly two dozen adventures we have of him, several show the sort of work that Morgan was doing during the War. So we know that these could have happened anytime.
Morgan's demonstrated experience and contacts throughout Europe, indicate that to him this is business as usual. That is also shown in the opening spiel given before each of the televised adventures we have of him:
"Secret File U.S.A - A warning to all enemies of America at home and abroad who are planning acts of aggression. This is the story of the gallant men and women who penetrated and are still penetrating enemy lines to get secret information necessary for the defense of the United States. This is the story of one our nations mightiest weapons. Past, present and future, if necessary - The American Intelligence Service."
While that statement talks of the "men and women" at work for the country's well-being, the one individual we see on each episode is Morgan.
We learn very little about Morgan's life outside of work. Considering how busy he is, he might not have much of a personal life.