First I am Catholic and nothing said here will offend me or take me away from my faith. I ask this because there did not seem to be a single authority or church for years after Jesus died, as far as I know the gospels were not written until at least 30 years after his death.
So how can the Catholic Church claim they were founded by St Peter or be more authentic than other churches? I know the religion lasted through various saints who were martyred but it doesn't seem until Constantine that the religion had one voice. Could it be argued that Constantine was the real founder of the Church and not St Peter?
Well, neither is probably the best answer.
The claim of the Roman Catholic Church is founded on basically two things: a documentary tradition that claims Peter was the leader of the church in Rome and passed on his authority to its bishop as a passing on of apostolic authority as well as primacy. However that documentary tradition is itself a development of the idea of Roman primacy that takes place over several hundreds of years. One cannot trace this idea, or at least the fullness of the claims of the RCC, back to surviving 2nd century documents, let alone 1st century ones.
It is complicated by the claim that there was always an unwritten tradition that accompanied the New Testament documents, an unwritten apostolic tradition that was authoritatively transmitted by the church, and that the RCC itself is the authoritative arbiter of such pronouncements.
You are right that the first Gospel text probably didn't appear until at least the 60s AD. Paul's earliest letters probably were written in the early 50s.
However Constantine is not really a good contender either. By the time Constantine emerges on the scene, there already exists a large, connected network of churches that recognise each other and have a concept of unity. Indeed, Constantine does not establish that at all, rather they appeal to him to deal with the problem of the Donatists, a separatist group dominant in North Africa. In that issue you already see a church that considers itself both unified and universal attempting to deal with a group of believers that the mainstream considers outside the church.
Furthermore, a very great deal of Constantine's dealings with the church take place in the Eastern half of the church, which down the track will comprise the portion that begins to be identified as Eastern Orthodox in contradistinction to the Roman Catholic Church dominant in the West.
Who founded the Roman Catholic Church is not actually that helpful an historical question to ask. As a tradition, the RCC can legitimately claim a historical continuity that goes back to the earliest period. But so can several other branches of Christianity. Furthermore, what exactly 'historical continuity' guarantees is not so certain. There was historical continuity between the Roman Empire in 1453 and the Roman Republic in 100 BC, but they were vastly different things. The RCC has a long and involved history in itself, and it's difficult even to decide when we should start talking about the RCC as something distinct from other 'Church' entities. Personally I wouldn't distinguish the Western church as "Roman Catholic" before the 5th century.