ANSWER: The phrase 'kingdom of God' is used more than sixty times in the King James New Testament, mostly in the gospels of Mark and Luke. A phrase that closely parallels it, which references heaven, is in the KJV thirty-three times. Interestingly, it is found exclusively in the book of Matthew.
A short answer to your question is that the Kingdom of God is the rule and governance of his Family over all things. The Bible speaks of it in the past (Daniel 4:17, 25, 34; 5:21), present (Matthew 12:28, Luke 17:20 - 21, Colossians 1:13), and future (Daniel 2:44 - 45, 7:13 - 14, Revelation 11:15 - 18, 20:4 - 15, 21:1 - 22:5). Right now, although it rules the entire universe, it is not actively governing over the affairs of man. God is, nevertheless, ruling in the lives of all those who have his Holy Spirit and are obedient to him.
In the future, it will be reinstituted on the earth at the Second Coming of Jesus as King of kings and Lord of lords.
44. And in the days of these kings, the God of heaven shall set up a kingdom which shall never be destroyed . . . and it shall stand forever (Daniel 2:44, HBFV throughout)
When Jesus returns all true Christians who have died or are still alive will be changed into immortal spirit beings. This transformation, known as the first resurrection, is necessary for the saints to enter into salvation and to rule on earth for 1,000 years (1Corinthians 15:50 - 53, see also 1Thessalonians 4:13 - 18, John 3:3 - 7).
After a thousand years are completed all those who never got a full chance at salvation will be resurrected to receive their opportunity to repent and live the way of God. The Father will then bring the New Jerusalem from heaven to the new earth and His government will rule the entire universe from the New Jerusalem for all eternity. The earth will literally become the center of all creation for ETERNITY.
Those who will be allowed into the Eternal's reign of righteousness must receive it like a little child (Mark 10:15), be doers of his will (Matthew 7:21 - 23) and keep his Commandments (Matthew 19:16 - 17; Revelation 22:14). They must be willing to seek it (Luke 12:31), to give up all things in order to possess it (Luke 18:29 - 30), love their fellow man (Matthew 25:31 - 46) and be willing to suffer persecution for its sake (Matthew 5:10).
What and who will NOT be allowed to enter the Kingdom of God? Those who are adulterers, those who practice sorcery, are contentious, have outbursts of anger, are selfish, those who promote false doctrine, those who cause divisions among believers, murderers and more (Galatians 5:19 - 21). The Apostle Paul also states in the book of Ephesians that fornicators and those who are covetous will not enter it either (Ephesians 5:5). Toward the end of the book of Revelation, we are also told that those who are cowards, perverts, traitors, those who are immoral, those who murder, those who practice magic and worship idols, and liars will not be in it. What they will be in, however, is the lake of fire (Revelation 21:8) whose sole purpose is to put to death all those who reject salvation.