For the ancients (Aristotle, Plato, etc...), there was no conflict inherent between church (religion) and those who governed. The rulers of the Greek city-state were guardians of the religious practices of their city; being a good citizen was being a good Greek was being a good religious believer.
With the coming of Christianity--and, to a more limited extent, Judaism--a sudden dichotomy appeared. Now there was the earthly kingdom and the Kingdom of God, the temporal powers and the eternal powers. This was a split made possible by Christ's radical call to "render unto Caesar what is Caesar's and unto God what is God's."
Philosophically, this was an enormous problem. Augustine tackles it in City of God. What is the relationship between the city of God and the city of man? What is a Christian to do, who is called to evangelize and serve on earth while belonging solely to the kingdom to come?
There is no such thing as "Christian politics" in the Bible. There is no civil law in the New Testament. The Christian does not see revelation as a revelation of law, but rather as a revelation of truth in the person of Christ. The only law is "to love," which does not translate into any one particular political program. (If you have difficulty finding a candidate to vote for in each election who does not support one nasty program or another, you know what I mean.)
The problem is that, in everyday life, the two kingdoms overlap. My obedience to the state and to the Church do come into conflict at times. Because of Christianity, political thinkers have an endless intellectual and practical problem to tackle (what fun!): how does one run a state in which its religious citizens have a divided allegiance? how do rulers engage or control citizens who care more about the next world than this world?
The two classic solutions are: (1) Build a state church. For example, the Anglican Church of England was a state church in which to be a good chruchman was to be a good Englishman. (2) Tolerate all religions equally.
Now the fun begins. Dead white guys and many more all jump into the fray to defend their favorite model or seek out the best model. And all because God became a man...