Mediatorial Kingship of Christ
I believe that I can say that no one can doubt the essential kingship of Christ. Christ, because he is the creator, is the essential king over all things because He has created them. However, an RPCNA distinctive is found in the doctrine and application of the mediatorial kingship of Christ.
The mediatorial kingship of Christ holds that Christ's kingship over all things in a special, somehow elevated way as the reward for His redemptive work. Matthew 4 shows us that that this was always that plan, as the final temptation that Satan sets before Christ is kingship without the cross; this is further confirmed by Psalm 2 and the promise by the Father to the Son of kingship over the nations. Christ Himself recognizes this in Matthew 28, speaking of all authority had been given to Him and using that authority to send the church throughout the world. This elevated rule is also seen the depiction of Christ in Revelation 19, which proclaims the glory and power of Christ, who is marked as the "King of king and Lord of lords".
However, oftentimes the unique applications of the doctrine within the RPCNA distinguished it from others who held similar views. The need for this doctrine to be applied flows from Psalm 2, as the rulers of the world are commanded to kiss the Son who has been set as king.
First, nations have a duty to recognize the rule of Christ over them. Nations that do not recognize the sovereignty and supremacy of Christ over them are in active rebellion against Him. On this basis, the early leaders of what grew into the RPCNA objected to the US Constitution as it failed to acknowledge Christ as King. For many years, the RPCNA was strongly involved in the Christian Amendment Movement to insert that recognition into the Constitution. On this point, the Testimony of the RPCNA states that:
Every nation ought to recognize the Divine institution of civil government, the sovereignty of God exercised by Jesus Christ, and its duty to rule the civil affairs of men in accordance with the will of God. It should enter into covenant with Christ and serve to advance His Kingdom on earth. The negligence of civil government in any of these particulars is sinful, makes the nation liable to the wrath of God, and threatens the continued existence of the government and nation.
Second, members of the RPCNA did not run for political office. Those elected to public office must take an oath to the Constitution, and it was not considered permissible to swear an oath to a document that is fundamentally in rebellion against Christ. Given the historical roots of the RPCNA in those who took swearing the National Covenant and Solemn League and Covenant, oaths - especially those related to the government - are not to be taken lightly nor in rebellion against God.
Third, members of the RPCNA did not vote. If members of the RPCNA were not willing to themselves take that oath, how could they ask someone to swear it on their behalf by their vote? This does not mean that RPCNA members were uninvolved in the political process (as seen in the Christian Amendment Movement); rather, they made their voices heard through other means than voting. However, members of the RPCNA who voted experienced church discipline because of that action. This position has changed within the last 50 years, as the official position of the RPCNA now states that:
The Christian, when such action involves no disloyalty to Christ, ought to be involved in the selection of and to vote for civil rulers who fear God, love truth and justice, hate evil, and are publicly committed to scriptural principles of civil government.
Normally, if you wanted to read more, I would recommend Messiah the Prince or Political Danger or an article by John Murray. However, I am glad I can now direct you towards Rick Gamble's The Whole Counsel of God, Volume 2, pages 475-523. Dr. Gamble might offer one of the most unique courses at a North American Reformed seminary with his "Doctrine of Civil Government", and there is a significant overlap between the content of the book and the content of that (excellent) class.