The short answer to this question is: we are not told. However, let me venture a reasonable suggestion.
Many of the great hymns and anthems of the OT (of which Judges 5 is one of the truly greats) were designed to be sung either by one person, a choir, two choirs, a soloist and and a choir, and almost every other variation. For example, Ps 106 & 107 appear to be antiphonal psalms for either two choirs, or, a soloist and and a choir. For example, see the musical arrangements made for the dedication of the wall in Neh 12:27-43 with two choirs. Musical instruments were also used in many occasions.
The content of the great anthem in Judges 5 simply recounts in beautiful Hebrew poetry, the history of events that had just occurred, including all the main players, such as Deborah, Barak, the princes of Israel, the LORD, Jael, Shamgar son of Anath, Heber the Kenite, etc, etc.
Just who sang what part of the song is not revealed. My personal preference for such a magnificent and grand hymn: there might have been two soloists and a choir? However, I assume it was a superb rendition.