During my years of training to become a Calvinist pastor, the issue of church polity was quite a live one. Calvinists themselves do not agree on what is the "biblical model" for church government. Presbyterian, Reformed, and Congregational denominations share a Calvinist doctrinal heritage but different governing structures. There was more or less a consensus that the New Testament was unclear about the manner in which church leaders should be selected and what their roles were.
It may be true that the New Testament leaves much unsaid about the role of church leaders, but I don't think it is as unclear as we thought it was. Rather, I think that what was clear was not seen by us, because it was unacceptable and unworkable for us.
I'm convinced that the New Testament shows a top-down Church governing structure in which each generation of leaders appoints the next, tracing back to the apostles. In other words, apostolic succession.
The principle of apostolic succession is that the leadership of the Church, by which we mean primarily, but not only, the bishops, were appointed by the previous generation of leaders, and they in turn by a previous generation, all the way back to the apostles, who appointed the Church’s first generation of leaders during their own lifetimes. Thus, the bishops are successors of the apostles in the sense that they fulfill the apostles’ role, which is one of leadership or oversight (episkope in Greek).
We see this pattern in Acts.
Acts 1:12-26, the replacement of Judas by Matthias, is significant. It does not prove apostolic succession. But it demonstrates two important points: (1) The apostles had a role or office, which did not necessarily cease with their death, (2) this role is described, among other things, as an episkopen, an “oversight” (“his office [espiskopen] let another take”, Acts 1:20 rsv). Calling the apostles’ role an episkopen shows the connection between the apostles and the later leaders of the Church, who are frequently called episkopoi (in English, “bishops”: Acts 2:28; Phil 1:1; 1 Tim 3:2; Titus 1:7).
In Acts 1, the church is growing already (120 people, Acts 1:15) and the apostles are short on leadership, because they are missing Judas. So he is replaced by Matthias. The apostles are back up to full strength of numbers.
In the beginning of the Church, they are able to perform all the roles of leadership, but this quickly becomes too much. They appoint more leaders (Acts 6:3), by the laying on of hands (Acts 6:60), to share the burden with them.
Later yet, the Church is going to spread all over the Mediterranean, to places the apostles cannot get to easily. Then, the apostles appoint other men to share in the “oversight” (episkopen). These men are called presbuteroi, “elders”, from which we get the English word “priest”: see Acts 14:23
In the beginning there is no distinction between presbuteroi and episkopoi: compare Acts 20:17 and 20:28. Later, these roles will be differentiated. It is like tissue in an unborn baby: at first the organs are one lump of cells, but they differentiate into different organs in time. In a similar way, the apostles had the role of bishop, priest, and deacon all wrapped in one, but these roles differentiate in time. All clergy share in Holy Orders and at least partially in apostolic succession, since they fulfill roles of leadership originally held by the apostles.
The leadership of the early Church was always appointed by the apostles, not elected. This pattern holds in Acts and also the Pastoral Epistles (see Titus 1:5). Even in exceptional cases, like Paul, who is made an apostle directly by Jesus, such a person goes to the apostles to receive confirmation (see Acts 9:27; Gal 1:18; 2:1-2, 9). If you ponder this principle of appointing leaders, you will see that it is a top-down structure, and it implies apostolic succession: all the church’s leaders, if legitimate, ought to be able to trace their appointment to the apostles, handed on in succession.
Look at Titus 1:5. Titus is Paul’s representative, his “child in faith” (1:4). Paul tells Titus to appoint presbuteroi and episkopoi (elders/priests and bishops, 1:5, 7) for Crete in every town. Appointing such people was something Paul used to do personally [Acts 14:23]. Now he’s passing the authority on to Titus. This shows us apostolic succession.
Look at Acts 20:28-37. Paul knows he is being taken away from the Church of Ephesus. He will no longer be able to lead them, due to imprisonment and ultimately death (20:29, 38). He passes them the torch of episkopen to them (20:28). Again, this shows us apostolic succession. The elders/overseers (in our usual terminology, priests/bishops) will guide the Church in Paul’s absence.