The dramatic words “the chains of hell” appear six times during the first third of Alma. While reflecting on their use in Alma 5, toward the end of Alma 12:11 Joseph Smith issues a clumsy passage to emphasize the captivity of the devil: “Now this is what is meant by the chains of hell.” It’s common to find such wording in the King James Bible, but it doesn’t occur there.
Since Alma is a fictional character, we’re left to determine if such a powerful phrase came from Joseph Smith, Jr. or another source of inspiration. It is quite possible the “chains of hell” was common ‘revival-speak’ in the day, since it sounds that way. The earliest reference I found, pre-dating the dictation of the Book of Mormon, is from the mid-1700s. Samuel Johnson translated the final line in Horace: Book IV. Ode 7 to read “The chains of Hell that hold his friend.”
Joseph Smith likely picked it up from a preacher, who may have heard it from another, who may have read this translation at some point, perhaps during his schooling for the ministry.
Further research into whatever documented preaching from the early 1800s is available would probably reveal many clues into the language of the Book of Mormon.