The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the apostles, Clement was allotted the bishopric.
Tertullian is often taken to imply that Clement was Peter's immediate successor, but Tertullian is sometimes a bit loose on historical matters and it's very possible that he didn't intend to imply anything quite so specific. Virtually everyone else takes Linus (who most take to have been consecrated by Peter, but the Apostolic Constitutions seems to regard as having been consecrated by Paul) to be Peter's immediate successor. The very influential (but late) Liber Pontificalis seems to me to suggest that Peter consecrated Linus, Anacletus, and Clement as bishops, but gave them different roles, perhaps with Linus and Anacletus being primarily concerned with the actual community at Rome and Clement being entrusted also with the relations between Rome and other sees. That, if true, would explain any confusion -- other sees would have known Clement, but may have been hazy about how the other two fit into the picture. But the Liber Pontificalis is somewhat obscure and muddled (it also treats Clement as Peter's immediate successor), so it's unclear how much weight it should be given in the earliest cases. There's a lot we don't know about how the early Roman church was organized, so we may just be missing a piece.
In any case, most lists put the order as Peter, Linus, Anacletus, Clement, and everyone holds Clement to have been consecrated by Peter. According to tradition, Clement was martyred under the Emperor Trajan, and today is his feast.
From his letter to the Corinthians (Chapter 42):
Christ therefore was sent forth by God, and the apostles by Christ. Both these appointments, then, were made in an orderly way, according to the will of God. Having therefore received their orders, and being fully assured by the resurrection of our Lord Jesus Christ, and established in the word of God, with full assurance of the Holy Ghost, they went forth proclaiming that the kingdom of God was at hand. And thus preaching through countries and cities, they appointed the first fruits [of their labours], having first proved them by the Spirit, to be bishops and deacons of those who should afterwards believe. Nor was this any new thing, since indeed many ages before it was written concerning bishops and deacons. For thus says the Scripture in a certain place, "I will appoint their bishops in righteousness, and their deacons in faith."