I'll assume you mean traditional catholic.
Well, let's start off with the first 3 branches of government; judicial, legislative, and executive. In order to join any of these branches, one would have to be a catholic. Freedom of religion concerning the political branch would be abolished, and in order to be eligible to run for a political position, he would have to be catholic. Catholics would also have more rights than people of other religions.
The judicial branch, I think, should be primarily made up of members of the clergy. I think that the Supreme Court should be made up of bishops. the lower branches of the judicial system should be two factions; clergy and secular, mostly to act as a sort of checks and balances. The clergy to uphold church law, the secular to hold up civil law. The Supreme Court of Bishops would see that congress was in accordance with Church teaching and constitutional law.
Congress would remain secular, but congressional meeting are to be done after a mass has been offered. The executive branch would also be secular, and can appoint bishops to the Supreme Court with the guidance of the pope.
The Constitution might have to be modified in some places. Freedom of speech would still be allowed but only to the extent that the speech is not against Church teachings. Freedom of the press, gun rights, right to vote, etc, can still be uphold. Women may or may not be allowed to vote.