What is the Kingdom of God?

First of all, we must shed the idea of kingdom as a realm or a people when trying to understand the kingdom of God.  It is first and foremost God's rule and action as George Eldon Ladd, a professor of New Testament Exegesis writes:

We must ask the most fundamental question: What is the meaning of " kingdom"? The modern answer to this question loses the key of meaning to this ancient Biblical truth. In our western idiom, a kingdom is primarily a realm over which a king exercises his authority. Not many kingdoms remain in our modern world with its democratic interests; but we think of the United Kingdom of Great Britain and Northern Ireland as the original group of countries which recognize the Queen as their sovereign. The dictionary follows this line of thought by giving as its first modern definition, "A state or monarchy the head of which is a king; dominion; realm."

The second meaning of a kingdom is the people belonging to a given realm. The Kingdom of Great Britain may be thought of as the citizens over whom the Queen exercises her rule, the subjects of her kingdom.

The exclusive application of either of these two ideas to the Biblical teaching of the Kingdom leads us astray from a correct understanding of the Biblical truth. The English dictionary itself makes this mistake when it gives as the theological definition of the kingdom, "The spiritual realm having God as its head." This definition cannot do justice to the verses which speak of the coming of the Kingdom in outward glory and power when Christ returns. On the other hand, those who begin with the idea of a future realm inaugurated by the return of Christ cannot do justice to the sayings about the Kingdom as a present spiritual reality.

Furthermore, those who begin with the idea of the Kingdom as a people base their definition upon the identity of the Kingdom with the Church, and for this there is very little scriptural warrant.

We must set aside our modern idiom if we are to understand Biblical terminology. At this point Webster's dictionary provides us with a clue when it gives as its first definition: "The rank, quality, state, or attributes of a king; royal authority; dominion; monarchy; kingship. Archaic." From the viewpoint of modern linguistic usage, this definition may be archaic; but it is precisely this archaism which is necessary to understand the ancient Biblical teaching. The primary meaning of both the Hebrew word malkuth in the Old Testament and of the Greek word basileia in the New Testament is the rank, authority and sovereignty exercised by a king. A basileia may indeed be a realm over which a sovereign exercises his authority; and it may be the people who belong to that realm and over whom authority is exercised; but these are secondary and derived meanings. First of all, a kingdom is the authority to rule, the sovereignty of the king. (The Gospel of the Kingdom. George Eldon Ladd. Wm. B. Eerdmans Pub. Co. Grand Rapids, MI. 1959. Pages 13-23.

Second, we find that the words associated with God's kingdom are not ones of building, bringing, or making it happen, but receiving, seeing, and entering it.  Again, Ladd speaks to this:

The supernatural character of the present Kingdom is confirmed by the words found in association with it. A number of verbs are used with the Kingdom itself as the subject.

The Kingdom can draw near to men (Matt. 3:2; 4:17; Mark 1:15; etc.); it can come (Matt. 6:10; Luke 17:20; etc.), arrive (Matt. 12:28), appear (Luke 19:11), be active (Matt. 11:12). God can give the Kingdom to men (Matt. 21:43; Luke 12:32), but men do not give the Kingdom to one another. Further, God can take the Kingdom away from men (Matt. 21:43), but men do not take it away from one another, although they can prevent others from entering it. Men can enter the Kingdom (Matt. 5:20; 7:21; Mark 9:47; 10:23; etc.), but they are never said to erect it or to build it. Men can receive the Kingdom (Mark 10:15; Luke 18:17), inherit it (Matt. 25:34), and possess it (Matt. 5:4), but they are never said to establish it. Men can reject the Kingdom, i.e., refuse to receive it (Luke 10:11) or enter it (Matt. 23:13), but they cannot destroy it. They can look for it (Luke 23:51), pray for its coming (Matt. 6:10), and seek it (Matt. 6:33; Luke 12:31), but they cannot bring it. Men may be in the Kingdom (Matt. 5:19; 8:11; Luke 13:29; etc.), but we are not told that the Kingdom grows. Men can do things for the sake of the Kingdom (Matt. 19:12; Luke 18:29) but they are not said to act upon the Kingdom itself. Men can preach the Kingdom (Matt. 10:7; Luke 10:9), but only God can give it to men (Luke 12:32)  (The Presence of the Future. George Eldon Ladd. Wm. B. Eerdmans Pub. Co. Grand Rapids, MI. Page 193.) (Italics mine.)

Dallas Willard has a good way of putting it:

The Kingdom of God is God reigning. It is present wherever what God wants done is done. It is the range of God’s effective will. God’s reign is all around you and is from “everlasting to everlasting” — it is the natural home of the soul. Matthew uses the term the Kingdom of the Heavens to emphasize that the Kingdom of God is not far off and way later but is immediately and directly accessible to us through Jesus Christ.

And so we find that Jesus's message of "Repent and believe the good news, the kingdom of heaven is at hand" is not about the kingdom that is about to happen or come, but one that is accessible now.  The good news is about how we can join Jesus in the kingdom through trusting him in our daily lives.