The Doctrine of Christ and the Gospel of Christ are similar, but how do they relate to each other? The doctrine is faith, repentance, baptism, receiving the Gift of the Holy Ghost and enduring to the end—those five points. And the Savior says, "And there will be no more doctrine." In other words, that is the doctrine of Christ. (See 3 Nephi 11:28–39).
But in 3 Nephi 27:21, the Savior says, "And this is my gospel." A few verses earlier, He stated, "This is the gospel which I have given unto you—that I came into the world to do the will of my Father, because my Father sent me. And my Father sent me that I might be lifted up upon the cross." So, one might say that when Book of Mormon writers use the word gospel, they may be speaking a little more broadly, referring to something we more often would call the Plan of Salvation. The specific doctrine of Christ is part of and wholly consistent with the Plan of Redemption, the Plan of Happiness. The great plan goes by several names, emphasizing parts of that plan. The terms doctrine of Christ and gospel of Christ, while overlapping considerably, may evoke somewhat different aspects of the same great Eternal Plan of the Father.
Book of Mormon Central, "What is the Doctrine of Christ? (2 Nephi 31:21)," KnoWhy 58 (March 21, 2016).
Noel B. Reynolds, "The True Points of My Doctrine," Journal of Book of Mormon Studies 5, no. 2 (1996): 26–56.
Noel B. Reynolds, "The Gospel of Jesus Christ as Taught by the Nephite Prophets," BYU Studies Quarterly 31, no. 3 (1991): 31–50.