We know Clement from the New Testament scripture, but Clement has not written anything about him being a Pope. I have never heard of any other early church fathers who says he was. How does the Catholic Church make this claim?
According to Our Sunday Visitor's Encyclopedia of Catholic History, He was either the successor of Cletus (aka St. Antecletus) or according to to St. Jerome and Tertullian was the immediate successor of St. Peter (and was consecrated by him).
... as also the church of Rome, which makes Clement to have been ordained in like manner by Peter
Clement, of whom the apostle Paul writing to the Philippians says "With Clement and others of my fellow-workers whose names are written in the book of life," the fourth bishop of Rome after Peter, if indeed the second was Linus and the third Anacletus, although most of the Latins think that Clement was second after the apostle.
So there was a discrepancy even back then - it could have seemed to Greeks that he was the Bishop of Rome, when there were actually two between.
Origen thought he was the Clement mentioned by Paul in Philemon 4:3.
His most famous epistle, the letter to Corinth is from the Church at Rome to the Church at Corinth. Might make sense for the Bishop of Rome to write a letter like that. St. Paul didn't write letters from "The Church at X" to "The Church at Y".
Allow me to answer your question with a question!
Question: Who was Pope after Saint Peter and how was he chosen?
Answer: Historians are generally in agreement that Saint Peter named his own successor, a practice not generally followed by future Popes, although one or two did try, most lists of papal succession give Saint Linus as Peter's immediate replacement, although the Liber Pontificalis (LP), has Peter consecrating Saints Linus, Cletus, and Clement as what we would today call "auxiliary" bishops, and naming Clement to replace him at his death. The same list has Saint Anacletus following Clement, but most modern historians feel that he was the same person as Saint Cletus (in spite of being born in a different city according to LP, and in spite of having a separate feast day in the Roman Missal until well into the twentieth century). The important thing, of course, was that the papal power was passed down in an orderly manner. -