Hobbes's basic argument for a National Church is quite easy to state: "they that are the Representants of a Christian People, are Representants of the Church: for a Church, and a Common-wealth of Christian People, are the same thing" (Leviathan, Chapter XLII; cp. Chapter XXXIII). In Hobbes's account of authority, authority involves personation or representation -- that is to say, for the community to act as a whole, its power to act must be invested in a person who can act for it. He holds, of course, that the authority of a commonwealth is personated in a Sovereign, making that person nothing less than the entire commonwealth in one person. But if church and commonwealth are the same people, then the church is already personated in the Sovereign, who is therefore Sovereign of the church as well as the commonwealth. This is why the book ends with an out-and-out attack on the Catholic account of the Church. As Hobbes sees it, Catholic ecclesiology is the diametrical opposite of this, since it makes of the Pope a Sovereign for the church, and therefore from the fact that the church and the commonwealth of Christian people are the same thing, claims for the Pope temporal power. But his aim is larger than this, since while he devotes his attention primarily to the Catholic position, scattered comments show that he also takes Calvinist ecclesiology to be an enemy. We see this, for instance, as he finishes up his criticism of Catholic ecclesiology (Chapter XLVII):
It was not therefore a very difficult matter, for Henry 8. by his Exorcisme; nor for Qu. Elizabeth by hers, to cast them out. But who knows that this Spirit of Rome, now gone out, and walking by Missions through the dry places of China, Japan, and the Indies, that yeeld him little fruit, may not return, or rather an Assembly of Spirits worse than he, enter, and inhabite this clean swept house, and make the End thereof worse than the Beginning? For it is not the Romane Clergy onely, that pretends the Kingdome of God to be of this World, and thereby to have a Power therein, distinct from that of the Civil State. From this consolidation of the Right Politique, and Ecclesiastique in Christian Soveraigns, it is evident, they have all manner of Power over their Subjects, that can be given to man, for the government of mens externall actions, both in Policy, and Religion; and may make such Laws, as themselves shall judge fittest, for the government of their own Subjects, both as they are the Common-wealth, and as they are the Church: for both State, and Church are the same men.
For Hobbes, as for Milton, new presbyter sometimes turns out to be just old priest writ large.
Sovereigns, therefore, are the supreme pastor of their national churches, with authority "to ordain what Pastors they please, to teach the Church, that is, to teach the People committed to their charge" (Chapter XLII). Hobbes takes this all the way, going so far as to argue that the Bible gets its authority for Christians due to its imposition by the Sovereign (Chapter XXXIII):
It is true, that God is the Soveraign of all Soveraigns; and therefore, when he speaks to any Subject, he ought to be obeyed, whatsoever any earthly Potentate command to the contrary. But the question is not of obedience to God, but of when, and what God hath said; which to Subjects that have no supernaturall revelation, cannot be known, but by that naturall reason, which guided them, for the obtaining of Peace and Justice, to obey the authority of their severall Common-wealths; that is to say, of their lawfull Soveraigns. According to this obligation, I can acknowledge no other Books of the Old Testament, to be Holy Scripture, but those which have been commanded to be acknowledged for such, by the Authority of the Church of England.
Reading Leviathan directly as an argument for the Church of England as a National Church is needed to make sense of the entire work in multiple ways. For instance, it clarifies what Hobbes is doing in his endless Scriptural exegesis throughout. It also explains the structure of the work, rising from general considerations of man, to the nature of commonwealth, to the notion of a Christian Commonwealth, and ending in a culminating attack on Catholic ecclesiology, the most extensive and obvious threat to the idea of the Church of England as a National Church governed by the Crown of England.