What does it mean to become like Christ?

St Peter says in his epistle that we are by grace what he is by nature. God is God by nature, capital G. We never participate in the Godhead. We never become capital G God. The best and most succinct way was put by St Athanasius the Great: God became man so that man might become a god, lowercase g. Christ is the God-man. When a believer enters into baptism and is imbued with the Holy Spirit, he too becomes a god-man, but that is the Holy Spirit dwelling within him.

It does not mean he becomes part of the Trinity. We must become like Christ, though we never become God and never reach the stature of God. We are progressing eternally into the likeness of God.