For Israel, God is the one true king and the hope of their nation - only Jehovah will reign. For them, "Thy Kingdom come" was a way to keep their hope alive for the day when God would come and judge His enemies and give salvation for His people, as He promised.
Jesus gives us 3 priorities: thy name, thy kingdom, and thy will. If we're not praying for the kingdom of God first, then our priorities are not aligned with God's plans.
John the Baptist's message was to repent because the Kingdom of God was near. Jesus primary message was that the Kingdom of God was at hand (the good news). Matthew 12:21-31.
When we pray this prayer, we must remember that His kingdom has already come (when Jesus was on Earth, He was the king and so His kingdom was there). Jesus addressed the false assumptions that people had about the kingdom of God through his parables. Pray that we may receive understanding about what His kingdom really is, and that He may be the king over and in us. In everything we say and do, we ask that God's influence will be known.
The kingdom of God has been inaugurated but Mark 25:34 shows that the kingdom hasn't been consummated (fullness of God's reign on Earth as it is in heaven). This is a prayer for the end of evil. The fact that the kingdom hasn't been fully realized is by God's grace and mercy for those outside of the kingdom to come in. This is a prayer to make us missionaries - to reach the lost so that they do not perish. Jesus warned us to expect a battle; there are spiritual forces trying to keep people out of the kingdom.