Thomas Campbell was the father of Alexander. Both were prominent leaders in what is known as "the Stone-Campbell Movement", aka "the Restoration Movement". (Barton Stone was at least as important to the history of the Church of Christ as were either of the Campbells, and perhaps more important.)
I wonder if Thomas Campbell was related to, or by teachings, to Alexander Campbell of "The Campbellites"? Many of Campbellites, from Sindey Rigdons group in Kirkland Ohio, merged with the early Church of Christ group when Joseph Smith had brought over some copies of the book of mormon in 1831. The Church of Christ had been the official name chosen by the first (The Bible+BoM) group comprised of only 6 members in Fayette New York April 6, 1830. Many, who did not follow The Church of Christ group on into deeper waters, as The Church of Latter-day Saints, and then futher onto The Church of Jesus Christ of Latter Day Saints, I think, remained The Church of Christ even is day. David Whitmer for instance. Eliza R. Snow, and her brother Lorenzo snow, from that early Campbellite group in Kirkland Ohio, remained with this first early group and on into the later LDS group as key presidential figures who's writings are still quite well read as "classic" by today's LDS members. We get materials from The Church of Christ I've noticed, that are used in official LDS Church historical compiliations.
In 1921, Sidney Rigdon, a Baptist, met Alexander Campbell and became a part of the Stone-Campbell movement as a rather prominent preacher. By about 1830 Rigdon began to hold and proclaim the idea that revelation was not complete, there was more to come. Some believe he already had contact with Joseph Smith. Alexander Campbell rebuked Rigdon for his peculiar ideas and, discredited, Rigdon joined up with Joseph Smith. It has been said, even by some among the Mormons, that a considerable amount of Joseph Smith's ideas in the Book of Mormon came from Rigdon.
The Campbells were not in any way favorable to the Book of Mormon, and Alexander was one of the first to expose it as a fraud. You can read his "Delusians, An Analysis of the Book of Mormon, With an Examination of its Internal and External Evidences, and a Refutation of its Pretences to Divine Authority" (1831) at the following link:
[AnalysisBookOf Mormon]
A very enlightening critique. After a review of the BOM, in his "Internal Evidences" section, Campbell noted the following:
This Joseph Smith overlooked in his impious fraud, and makes his hero Lehi spring from Joseph. And just as soon as his sons return with the roll of his lineage, ascertaining that he was of the tribe of Joseph, he and his sons acceptably 'offer sacrifices and burnt offerings to the Lord.' - p.15. Also it is repeated, p. 18 - Nephi became chief artificer, ship-builder and mariner; was scribe, prophet, priest and king unto his own people, and 'consecrated Jacob and Joseph, the sons of his father, priests to God and teachers - almost six hundred years before the fulness of the times of the Jewish economy was completed. p.72. Nephi represents himself withal as 'under the law of Moses,' p. 105. They build a temple in the new world, and in 55 years after they leave Jerusalem, make a new priesthood which God approbates. A high priest is also consecrated, and yet they are all the while 'teaching the law of Moses, and exhorting the people to keep it! - p.146,209. Thus God is represented as instituting, approbating and blessing a new priesthood from the tribe of Joseph, concerning which Moses gave no commandment concerning priesthood. Although God had promised in the law of Moses, that if any man, not of the tribe and family of Levi and Aaron, should approach the office of priest, he would surely die; he is represented by Smith as blessing, approbating, and sustaining another //12// family in this approbated office. The God of Abraham or Joseph Smith must then be a liar!! And who will hesitate to pronounce him an imposter? This lie runs through his records for the first six hundred years of his story.