In the novelization of Rambo: First Blood Part II, John mentions that his father drank and beat his mother, Marie Drago, but this was not mentioned in the film canon. R. Rambo also tried to kill John with a knife at one point during a drunken rage, but Rambo defended himself by shooting his Compound Bow at his father, badly injuring and nearly killing him. Rambo would run away that night and later be drafted into the army. The result of this fight led to the two men not speaking for over forty years.
R. Rambo's first name is unknown. Sylvester Stallone has joked that it is "Roberto", which could be a reference the Rocky series of films, which also star Stallone, Rocky's real name is "Robert". But it is believed that his first initial is 'R' because of Richard Crenna, who the film is dedicated to, as his character has been seen as a father figure to Rambo.
We do not even know if the character is still alive, but if there is another film, we will probably be given more information on his character.