Johnny Fontaine was envisioned by Coppola as a young Sinatra, and was played by singer Al Martino. Martino didn't have much acting experience and was having difficulties getting his expressions and emotions right when the camera was on him. So, when he is in the office with the Godfather, the reason the camera is so often at his back is because he felt uncomfortable and unable to do the scene with the camera focused directly on him.
During the shoot of the movie, many people were unhappy about the quality of acting Pacino was giving. They thought he was showing the character as dumb and slow-witted. It wasn't until the scene of the Sollozo murder, on the third and fourth day of shooting, that the big heads at Paramount saw "quality acting" on Pacino's part. However, Pacino was still not highly regarded until the Godfather became a big hit, and Coppola was criticized immensely, and was threatened to be fired for his cast choices and the manner in which he was filming the movie.
In the scene outside the hospital when Michael encounters Capt. McCluskey, the officer standing to the left of McCluskey, "Phil" is actually former legendary New York City police officer, Sonny Grosso. Grosso was the actual partner of Eddie Eagan, who was portrayed by Gene Hackman as Popeye Doyle in "The French Connection" (1971). Grosso's character, Buddy "Cloudy" Russo was played by Roy Scheider. Grosso went on to direct and produce numerous police dramas for TV (the best known is "NYPD Blue"), usually about the New York Police Department.
Originally the character of Connie was supposed to be played by someone Coppola called "plain looking, the daughter of a big-shot who is only married off because she's the daughter of some big Mafioso guy". When they couldn't find an actress, Talia Shire, Coppola's sister, got the part, even though Coppola thought she was too beautiful and said "C'mon, look at her. Who wouldn't want to marry her?"
The little "game" Vito Corleone is playing with his grandson, with putting oranges in his mouth to look scary, is actually a "game" Marlon Brando plays with his real grandchildren.
In the scene where Michael is rolling Don Corleone on the hospital gurney he accidentally mashes the Don's hand between the gurney and the doorway. Marlon Brando snatches his hand away, Al Pacino briefly pauses, then continues with the shot.
When Luca Brasi goes to visit the Tattaglia's and is strangled, you can see his face turning slightly black due to strangulation. This effect was achieved by placing a type of translucent powder on the actor's face which tints black when it comes in contact with water. So while Luca Brasi was being strangled, a fine mist of water was sprayed over his face to trigger the colour change.