Virtually all religious groups that align themselves with Christianity make claims about their connection to the early Church. These claims can generally be divided into types – those that claim continuation with the early Church, and those who claim to be restoring the faith to it’s early roots.
The first question is, what is meant by the early roots. Almost always this refers to the Church of the New Testament era, and generally not a day later. Some have asserted that the Church began to fall away almost immediately after the death of St. John the Evangelist. The Anglican Church, however, has tended to make the argument that the Church continued on for a period after the repose of St. John, but later fell into error. More recently, I’ve seen writings by modern groups (most notably Reformed Baptists) that seem to imply that they, too, see the Fathers of the post-apostolic age as continuing on in the doctrine to which these groups adhere. The one thing that all of these groups have in common is that doctrine is the entire definition of the Church. That is, any group holding to the same doctrines as the early Church, are thereby members of that same Church.
Of course, “holding to the same doctrines” is a bit of a tricky question. One has to determine what those doctrines are. To some, all doctrines are contained within the covers of their 66 book Bible. Others, as I mentioned, believe that these doctrines are contained both in Scripture and in the writings of the early Church. There is little effective difference between these two views. In both cases, one needs to come to the text with an interpretive framework that helps you understand the text, and deal with those parts which are either unclear or appear to be contradictory. In both cases, as well, there is no foundation in either Scripture or the writings of the Fathers to hold to the belief. St. Paul, himself, clearly refers to teachings that are not contained solely in his letters, but that were transmitted orally. These teachings are part of the παραδοσισ of the Church. The word, often translated as tradition, refers to that which is handed down. So, the teachings of the Church are part of a larger body of knowledge that has been transmitted, or handed down, throughout the history of the Church. Yes, this includes Scripture, and the Church Fathers, but it also includes the prayers, the hymns, and also an oral tradition.
However, the Church has a much different view of itself. We find that view in St. Paul’s epistles, where the Church is described as the very body of Christ. This is not simply some rhetorical device, its a statement of fact. There is an organic wholeness to the Church. It is certainly the case that holding the same beliefs is a key component to what unites us, but there is also something more. That something more is Baptism and Chrismation. It is through this mystery (the two are done together in the ancient Church, and Orthodoxy has preserved this), that we are united to Christ, and therefore united to the Church. In the traditional forms of the Baptism service, either the Godparent or the new member are asked to state three times that they unite themselves to Christ. This, then, points to another aspect of being united to the Church. It is through our will (or those appointed to speak for us), and therefore we must have the capacity to depart as well. Historically, this, of course, has happened, and the Church has had to decide how to bring those who have left back in, if they so desire. Frequently, this is through Chrismation, as the Church has always been very concerned to not baptize anyone more than once.
So, the only way to be truly connected to the early Church is doctrinally and temporally – that is, being physically and spiritually part of that body which has existed since Christ established it. Realistically, any other approach is simply man made.