Edit: I realize the question is broad. I'd be happy to hear even about particular eras, whether late antique, medieval, renaissance, modern, etc.
Catholics have had mixed feelings on Greek philosophy and poetry. On the one hand, they certainly acknowledge a significant harmony between Platonic and Stoic thought and Christianity. On the other hand, there is a caution to avoid a preference for Greek philosophy over or against Scripture. I would say less that it evolved and more that it ebbs and flows.
St. Justin Martyr, writing for a non-Jewish audience, introduced the divinity of Christ by connecting the Logos of The Gospel of John, Chapter 1 with the Stoic Logos.
St. Clement of Alexandria's view on Greek philosophy is perhaps a high-water mark on Christian views re Greek philosophy: " "Philosophy has been given to the Greeks as their own kind of Covenant, their foundation for the philosophy of Christ ... the philosophy of the Greeks ... contains the basic elements of that genuine and perfect knowledge which is higher than human ... even upon those spiritual objects."
True, St. Augustine argues against pagan philosophy in The City of God. Yet he also notes that he found Cicero and Neoplatonic writings helpful on his journey towards the Catholic faith.
Boethius wrote his Consolation of Philosophy in the manner of a Platonic dialogue. Henry Chadwick, specialist on early Christian philosophy, describes the Consolation as "a work written by a Platonist who is also a Christian." Boethius also translated a number of Platonic and Aristotelian works prior to his execution.
During the fall of the western Roman Empire, classical Latin literature became less appreciated. St. Gregory the Great, for example, taught that bishops ought not to read pagan literature. St. Martin de Tours said that men had better things to do than study. Nonetheless, many monks, inspired by Cassiodorus, copied out many Latin texts. The following is a list of Latin texts where the oldest known copy (or most complete copy) was copied by a medieval monk:
The influence of Aristotle on St. Thomas Aquinas goes without saying, yet Aquinas disagreed with him in a number of places.
Dante, of course, was an admirer of Virgil and other classical figures.
In the sixteenth century, when Greek and Latin classics were quite popular among the literate class, the Jesuits often taught their students Latin using Cicero.
This is, of course, an incomplete overview of the relationship between Christianity and the classics. Yet one can see that there is always at least some appreciation, even if it is not uncritical or prominent.
St. Augustine of Hippo, Confessions
------, City of God
St. Clement of Alexandria, Stromata
St. Justin Martyr, First Apology of Justin Martyr
Boethius, The Consolation of Philosophy
Dante Alighieri, The Divine Comedy
Henry Chadwick, Boethius: The Consolations of Music, Logic, Theology and Philosophy, 1990
Cook, "How Deep the Platonism," 269-286 in Farms Review of Books, vol. 11, no. 2 (1999).
Lejay, Paul. "Classical Latin Literature in the Church." The Catholic Encyclopedia. Vol. 9. New York: Robert Appleton Company, 1910. 11 Mar. 2021 http://www.newadvent.org/cathen/09032a.htm.