Not in his lifetime.
He told Peter that he (Peter) was the rock upon which the church would be built, but the building of the church was probably more the work of the Holy Spirit at, and following, Pentecost.
No-one specifically. He told Peter that He (Jesus) would build the Church.
Restored Church of Jesus Christ was created in 1980.
Jesus said you are Simon I will call you Peter. On you I will build my church.
Are you asking about the Church of Jesus Christ of Latter-day Saints or the Church of Jesus Christ? The Church of Jesus Christ is another name for Christianity, and was obviously founded by Jesus Christ. There may of course be individual Christian churches with the name "The Church of Jesus Christ", which were probably started by the pastors or parishoners. The Church of Jesus Christ of Latter-day Saints is a Christian denomination, it is founded on the teachings of Jesus Christ but was started by Joseph Smith.
He founded it, and is its Head. The "Church" is now the 'earthly' body of Christ. *Matthew 16:18 - And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it.
Our Lord Jesus Christ, established the Christian church, when he made St Peter the Rock upon which he would build his Church.
Church of Jesus Christ - Drewite - was created in 1965.
Church of the Lord Jesus Christ was created in 1933.
Church of Jesus Christ - Toneyite - was created in 1980.
Holy Church of Jesus Christ was created in 1964.
Church of Jesus Christ in Madagascar was created in 1968.
Church of Jesus Christ - Cutlerite - was created in 1853.