Derek Thomas in Praying the Saviour's Way, gives the following explanation of "Thy kingdom come":
First, this petition alludes to the sovereign rule of God as King over the entire universe. The Lord who merely speaks all things into existence at the creation is King. His word is authoritative and powerful.
Second, this petition alludes to the covenantal rule of God over his people... On every page of the Old Testament there is the expectation that God is working out a plan and purpose in which he is gathering a people to himself and over which he intends to exercise his rule...The church of the New Testament is the gathered people, the seed of Abraham: 'If you belong to Christ, then you are Abraham's seed, and heirs according to the promise' (Gal. 3:29).
Third, this petition alludes to God's intention to overthrow all of Satan's pretension to power...'The reason the Son of God appeared was to destroy the devil's work' (1 John 3:8). 'And having disarmed the powers and authorities, he made a public spectacle of them, triumphing over them by the cross' (Colossians 2:15).
Fourth, this petition alludes to the as yet incomplete nature of the kingdom of God. We live as Christians...in the time between the two great advents of Christ. The Incarnation is past; the Second Coming is future. The kingdom of Christ has come and the kingdom of God is yet to come! There is 'now,' but there is also a 'not yet'...The decisive battle has been won, but the ultimate victory celebration must await the final triumph of Christ in the establishment of the new heavens and new earth...In personal terms, this means that although the decisive change has taken place in our regeneration and union with Christ (we are not, nor can ever be, what we once were), the change is incomplete. We are sinners still, and hence we feel the pull of sin that would (if it could) drag us down so as to deny Christ entirely. We wrestle, then, against the world, the flesh and devil and cry out for deliverance.
Praying this petition of the Lord's Prayer, then, has in view the ultimate triumph of Christ in the gathering of the church, as well as the visible defeat of Satan in our own lives as we struggle with ongoing sin. Every victory against sin and Satan is an advancement for the kingdom of God.