When you think of the word kingdom, what does it bring to mind? A form of government, right? Yes. Historically, there have been all forms of governments: democracies, oligarchies, plutocracies, monarchies, totalitarianism, and so on. God's kingdom is just how it reads - a theocratic government with Christ as king. So why pray for God's kingdom?
It is the only actual government that will bring relief and justice to all of its citizens. Where is it coming? The next words in the prayer will help with that. "Let your will take place on earth as in heaven." Do we imagine that heaven is in chaos like our home here on earth? Certainly not. So the prayer is that by means of God's kingdom government - with Christ as king - God's will for a peaceful and secure life will take place on earth just as it does in heaven.
Well in order for earth to be like heaven, what's going to have to happen down here? Going to have to be cleaned up, right? Right. That is what God's kingdom is going to do: clean up earth, get rid of the troublemakers while protecting those who want to be citizens of His government, and then usher in a completely new system of governance that will supply everything humans are clamoring for now, beautiful homes, decent jobs, plenty food, safety and security for themselves and their families, justice, freedom from war, threat of war, crime and terrorism. Is that a utopian dream? Perhaps for some. But think about it, isn't that what leaders promise now? Isn't that what they fight for now? Don't they promise that if you put them in the proper leadership positions, they will deliver that to all? Yes, that is the promise. That is what all decent people want. But it is what only God's government can, and will, ultimately provide.