"When did the church of Christ begin?"
As already mentioned, the church of Christ is not a denomination founded by man nor a part of any religious organization founded by man. It originated in Heaven in the mind of God and was set up on earth in the city of Jerusalem on the first Pentecost following the resurrection of Jesus Christ, the Lord (Ephesians 3:8-11; Acts 2).
Instead of working with denominations, members of the church of Christ plead that all believers in Jesus Christ heed His prayer for unity and become one in His blood bought body, the church (John 17:20-21; Ephesians 4:4-6; Acts 20:28). Some might call that non-denominational or anti-denominational; we call it Biblical.
The foundation of the church of Christ is Jesus Christ Himself (1 Corinthians 3:11). A building is only as strong as its foundation. The same is true of a church. If a church is built upon a man or a human creed, it will fall. Only Jesus Christ has a right to build a church for He alone is “the Christ, the Son of the Living God” (Matthew 16:13-19).