In the novels
The original novel explains Tommy Neri's backstory. He was born to Rita Neri and an unnamed father in an apartment in the Bronx. He grew up in the same neighborhood as Peter Clemenza. He was also frequently beat by his uncle, Al Neri, for his disrespect toward his mother. He never accepted that his uncle was a cop, and became a criminal just to spite him. He also wanted his 'button' but was told to wait until the right time.
In 1955, he became a 'made man' in the Corleone family under Michael Corleone at a hotel/casino ballroom owned by the family, with others. He was placed in the regime of Rocco Lampone and served him faithfully. The later novels explain he also becomes a heroin addict around this time. He is placed in Ritchie Nobilio's regime after Rocco Lampone is killed in 1959.
In The Godfather Returns, he is placed in charge of finding Nick Geraci. He makes no significant progress with finding Geraci, although he does look in Cleveland and gets tips from the C.I.A. He and his uncle Al torture and kill Nick Geraci's father.