To what extent has Christianity "appropriated" from Greek philosophy, and what did it borrow?

"Appropriated" has a broad enough definition to make it work in the broad sense, but in the narrower common-use sense of taking something alien and making it one's own, suggesting that a population like that of 1st century Christianity appropriated anything culturally Hellenic is mostly false and obscures the reality.

Early Christianity was a product of continuous mutual cultural exchange and simultaneous co-evolution within a largely Hellenistic cultural sphere which spoke Greek as their common language (in addition to other local languages like Syriac or Aramaic). Its population included cultural and ethnic Greeks and lived in a region already dominated by Greek culture. It emerged in the context of multiple non-Christian 1st and 2nd century philosophical and religious movements in the world trade and cultural nexus that was the eastern Mediterranean. It was in continuous conversation, exchange, debate, and integration with, and schism into, those non-Christian or semi-Christian elements.

In parts of Western Europe, later Christianity re-integrated locally extinct or dormant philosophical textual traditions, but since those had been present and prevalent at its formation among the people and ideas from which it was initially constituted, even calling that "appropriation" is odd.

Something more worth calling "appropriation" is the treatment of specific authors. Early medieval Christianity, especially in the West, worked out the theology of how to interact with non-Christian intellectual authorities by categorizing them as a sort of quasi-pre-Christians, communicating valuable truths that Christianity could place in proper context. The theory was that persons of good will could imperfectly grasp partial truths, making it not so different from modern ways of looking at knowledge, but framed in the context of a belief in the absolute truth of divine revelation.

In brief: Christianity can be said to have "appropriated" Greek philosophy to the extent that jazz could be said to have "appropriated" ragtime: literally true for some meanings of the word, but misleading, since it was an overlapping population, in the same cultural context, developing the same ideas, and sometimes the same actual human beings doing it.

Christian Medieval theology is in great part based on Aristotelian philosophy

The very same idea of "Theo-logy": a rational study of God, is due to the deep influence of Greek Philosophy.

The weird idea to prove God's existence was not a "necessity" of religious belief, but was due to a sort of "scientific" approach to God.

See Medieval Philosophy for an overview and history.

Early christianity - before christianity became a state religion and came to dominate the world view of people living around the Mediterranean Sea - shows various, conflicting ways to translate, explain or justify Christian ideas into Greek philosophical terms.

The most important issue was how to express the relation of Jesus to God. The general answer was something like: Jesus was somehow both human and divine. But how? What did this mean? Jesus is the Son, the Christ - okay, and what does that mean? To grapple with this (and explain it to non-Jewish heathens) the early church fathers had no other way than to use the terminology of Greek philosophy (and doing so, also create new meanings). So, in the Nicene Creed (325 AD) the relation of Jesus to God, the Father, is as expressed as "homo-ousios", "consubstantiality" ("three 'persons', one substance"). One conflicting view, that didn't make it, was Arius' view that Jesus, the Son, was a created creature, lacking the eternal divinity of the Father. The word 'persons' in this context is itself a kind of compromise between the Latin "persona" (originally meaning "mask" or "actor") and Greek "hypostasis" (which had originally the same meaning as "ousia": "what underlies something", "its essential being").

In some sense the Nicene Creed, which is what almost all Christian groups accept as their defining creed, is like a monster of Frankenstein: an amalgam of originally Jewish religious terms (like "Christ", God as creator, as judge) partially expressed and codified in Roman/Greek philosophical terms. (But actually, this translation effort is already happening also in the letters of Paul. For instance in his use of the term for "conscience", "syneidesis", which originated in Stoic philosophy; Paul gives it his own theological twist, of course.)

What is interesting in early christianity is the rich variety of conflicting "philosophical" views (like Arianism, Nestorianism, but there were lots of others - all duly anathemized; some, like Nestorianism, surviving for quite some time outside the orbit of the Roman empire). As a heathen I have to wonder: Can we draw any conclusion from the fact that practically every possible combination of views was adopted by this or that church father? (See wikipedia's schematic overview of christological views.)