Greek became the lingua franca of the wider region well before the birth of Jesus and continued in that status for some time (crudely 300 BC to 600 AD in the eastern Mediterranean). Even many Hebrews used Greek in the daily life and could not understand Hebrew language. It is natural that any writings targeted to a wider audience would be written in the lingua franca.
That does not mean that the use of Greek language would have meant adopting the Greek worldview, although some part of the Hebrews adopted partly the Greek lifestyle and the Aleksandrian Jews were fans of Platonism. I guess it tells something that the Christians mainly used Koine Greek (lingua franca), not the Classical Greek that the upper classes and learned people used.
As long as the leadership of the new movement was in the hands of persons having Hebrew background, Hebrew thinking permeated the teaching despite the Greek language. Greek philosophy probably started to play a stronger role after the leadership was switched to persons with roots in the Greek-Roman culture. The transfer of leadership started during the first century, so the Hellenisms in teaching may have almost as long history as Christianity.
As @St.Roymond has explained, the roots of Trinity can be found from the Hebrew scriptures and theology. Incarnation also fits to the same background, although it may have been more easy to accept for those with Pagan background. In that sense, I do not agree that these doctrines are “absolutely steeped in Greek Philosophy”.
[I rewrote the ‘soul’ part]
The concept of the ‘soul’ developed and partly changed among Hebrews during the long history from the times of Genesis to Jesus.
The Hebrew worldview included the underworld, Sheol, where the dead were supposed to go, an underworld place with stillness and darkness. Sheol was apparently used both as a generic metaphor of grave/death, and as a word meaning an actual realm of afterlife, so it is not always obvious how the word Sheol should be interpreted. Even the schools of Jews (Sadducees and Pharisees) disagreed about what Sheol is, just a grave or pit below the ground vs. a realm where the deceased are. Whatever it was, all would end there, both good and bad.
Apparently the early use of the word associated Sheol to a ‘family grave’-type endpoint. During the Second Temple period, there came input from various directions and the ideas of Sheol changed and diversified. Sheol had been the common endpoint for both good and bad but during the Second Temple period, many adopted the belief that ‘good’ and ‘bad’ had different destinies. Even Jesus seemed to teach this kind of ‘rewards or punishment after death’ type afterlife.
Whatever we think of Sheol, the texts in the Hebrew Bible reflect a holistic understanding of humans. Humans did not have souls, humans were souls (living, breathing beings). How that is related to the possible existence after death remained an unanswered question. If there was a resurrection, it was resurrection and life as a holistic body.
In the NT, the Letters of Paul speak about soul and spirit. These expressions can be understood as references to the different capabilities or ‘dimensions’ of a holistic human, like the texts speak about the ‘heart’ of humans. This is a matter of interpretation, so I understand that some try to use these expressions as evidence for a separate soul and spirit in humans (2-3 separate parts).
My current interpretation is that we are souls, instead of beings having souls. As I wrote, I leave this interpretation open to alternatives because I do not know certainly, so I do not claim that we do not have a separate, eternal soul. Basically, I support a ‘nephesh’-type holistic thinking rather than the input of the Greek philosophy.