Despite his reputation as a literary recluse, the collection - at the University of East Anglia - shows he also enjoyed trips to Niagara Falls.
It consists of 50 letters and four postcards written to the late Donald Hartog, from London, from 1986 to 2002.
They have been made public to mark the first anniversary of Salinger's death.
He died on 27 January last year at the age of 91.
In the letters, the American author said he hoped that when Henman faced German opponents at Wimbledon he "knocked 'em all down".
He also praised Henman's mother and father, saying they were not archetypal "professional tennis parents".
And he told his friend Mr Hartog he preferred Burger King burgers to those from other chains because they were  flame-grilled, making them "better than just edible".
Mr Hartog's daughter Frances, who donated the collection to the university, said: "There is tremendous warmth and affection towards my father and this is so different to the man Salinger is often portrayed as.
"The letters have been sitting in a drawer but hopefully, by being in the archive, they will show people another side to him."
Mr Hartog and Salinger met as teenagers in Vienna where they had been sent by their families to learn German.
"I think there was this extra bond between my father and Salinger because they met before the war," Ms Hartog added.
In March last year, another collection of unseen letters written by Salinger - to friend and former neighbour E Michael Mitchell - was displayed at a New York museum.
JD Salinger was a fan of tennis player Tim Henman and enjoyed eating burgers, previously unseen letters written by the Catcher in the Rye author show.