According to Catholicism, how did Clement of Rome become a Pope? 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
Tertullian - Against Heresies, Chapter 32

Jerome says

Clement, of whom the apostle Paul writing to the Philippians says &quot;With Clement and others of my fellow-workers whose names are written in the book of life,&quot; 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.
St. Jerome - On Illustrious Men

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 &quot;The Church at X&quot; to &quot;The Church at Y&quot;.