This passage occurs in the RCL "Pentecost"/"Ordinary"/"Proper" Season, Year A, most recently October 2014.
Summary: One does not find the Greek words for church and state in this passage, as much as this passage is used to justify all sorts of behavior and relationships between church and state. What is mentioned though is the word "εικον" meaning icon, or image. The temptors of Jesus, forgetting Genesis 1, say that the coin bears the image of Caesar. They answer the truth, but not the whole truth. An image of man is still an image of God. Money, whether it says, "In God We Trust" or "Caesar" or anything, isn't exempt from God's creation. It still has to do with humans and how we live in this creation, and thus it still belongs under God's dominion.
Freedom note: I am using this passage to launch a Reformation 500 series on the Freedom of a Christian. I pick this passage because Jesus discusses that even those of us free in Christ still have responsibilities before other people.
παγις ("hunter's trap", used as a verb, 22.15) The word for ensnare comes from the root for trap. What a cruel image of the pharisees trying with metal jaws, to trap Jesus.
αποστελλω ("send" 22.16). The literal phrase here is that his enemies "apostled their disciples," a reminder that Jesus is not the only one with apostles and disciples...
υποκριτης ("actor/hypocrite", 22.18) The word for hypocrite means actor, or one who plays a part. (He answered above the others from stage.) This is not necessarily a negative word, but in the NT it is used exclusively that way. Jesus isn't interested in actors, but real people with real sins that need real forgiveness.
εικον (image/icon, 22.20) The word here for "head" or "portrait" here is literally "eikon," which means image. So the question is whose image? If it is a human head, the answer could just as easily have been "God." (See Genesis 1!)
Translation/Grammar review: Idioms
Some things in a language are simply impossible to translate literally. This week Jesus is told, "You do not look into the face of people." This doesn't sound so nice. It simply means, "You don't look at exterior things." (Which is a positive assessment). He is also told he doesn't care about nothing. Missing from this idiom is the word "opinion." Jesus doesn't care about the opinions of others, in the sense that he acts free from petty judgments of others. You could take them literally, and perhaps derive some meaning; that said, with idioms, it is often best to let professional translators do the work...