Here are some comments from Eric Osborn, a patristic scholar who specialized in the study of Clement:
"Clement gives us insight into the Christian community at Alexandria, which has its orders of deacons, priests and bishops but is not set under one supreme bishop. Teaching is the chief activity of the church; its authority comes from a succession of teachers, not of bishops. Clement never speaks of an episcopal succession in Alexandria, but shows no sign of tension between priests and teachers. He writes (22.214.171.124) that 'he is a true priest of the church and a true deacon of the will of God, who does and teaches the things of the lord; for it is not because men have obeyed him and he is a priest that he is considered righteous; but it is because he is righteous that he is inscribed in the priesthood'. Authority is shared within the church between teachers and priests; later tension develops when Demetrius becomes the supreme bishop and the conflict causes the departure of Origen from Alexandria….Telfer comments on Clement's identification of the true presbyter with his true gnostic (126.96.36.199). 'Like Polycrates, Clement believed that whoever discharges the office of a bishop will receive corresponding exaltation in heaven. But on earth he rejoiced to see the apostolic faith coming to life again in the person of great Christians, rather than as being dependent for its preservation upon a succession of mediocre ones.'…The sequence here in Clement's mind is from logical exposition of scripture to authenticity of the knowledge in the church. He does not argue that an opinion held by the church has to be the most accurate, but rather that the most accurate and logical interpretation leads to the church….when he discusses heresy, there is little account of church discipline. The emphasis is always upon reason and the way in which scripture should be interpreted. This is the decisive factor." (Clement Of Alexandria [New York: Cambridge University Press, 2008], 22, n. 17 on 22, 216-7, 225)
In another part of the book, Osborn gives many examples of authorities Clement appeals to in his writings, such as teachers, the apostles, Jesus, and Christian tradition (175-6). A papacy isn't among them. Similarly, a papacy never comes up in other relevant contexts, like discussions of the church and unity (213-6).
Regarding the afterlife, Osborn mentions that Clement only gives us "the beginning of the idea of purgatory but does not anticipate its later developments" (52). For more about how Clement, as well as his Alexandrian contemporary, Origen, viewed the afterlife, see here.
In previous threads, I've given other examples of how Clement differed from Catholicism. See here on Mary in the Bible and here on whether Mary was sinless. On prayer to the dead, use the Ctrl F feature on your keyboard to search for "Clement" in the thread here. See this concerning the veneration of images.