Chains was written by the Brill Building husband and wife team of Gerry Coffin and Carole King - yes, that Carole King. The Beatles played several of that legendary team's numbers in their live act - you can hear them do Keep Your Hands Off My Baby and Don't Ever Change on Live at the BBC - but this was the only one that they would record as part of their official catalog. As songwriters, Lennon and McCartney were great admirers of Goffin and King, and stated so in interviews.
It was George Harrison, however, who saw this song as a vehicle for himself and brought it to the attention of the Beatles. The liner notes for On Air - Live at the BBC Volume 2 reveal that George bought the single at manager Brian Epstein's NEMS record store in December of 1962. The group quickly learned the number and put it into their act, soon performing it at the BBC on January 16th as you can hear on the Volume 2 collection.
They did four takes of the song on February 11th, though it turned out that take one had been the best. Like most of the recordings made on this day, they simply played it live in the studio with no overdubs. One of the only additions that they had made to the Cookies' arrangement was a harmonica flourish by John during the introduction, which he manages to get in just before he has to begin singing (he does not play the harmonica on the BBC version).
Though George is the featured singer in the bridges, most of the song is sung in three part harmony by George, John and Paul. This was, of course, one of the group's strong suits over their career, but this was the first time that their natural vocal blend was heard on record. This would also be the first time that listeners would get to hear George's voice up front. His other lead vocal on Do You Want to Know a Secret had been recorded earlier in the day, but it would appear later in the running order on the album. His voice is already more assured on Chains, even though it is a bit rough after a long day of singing backing vocals on several other numbers.
The song remained in the group's repertoire throughout the rest of 1963. They wound up playing it three more times for the BBC.