In what ways was the Byzantine Empire influenced by Christian and ancient Greek teachings and culture?

Answer 1

Constantine did not accept Christianity until he was on his deathbed. He liked it, and legalized it, but preferred his pagan beliefs.
To the point that some of the pagan belief would be blended into Christianity. The trinity doctrine is one of them.

The Greek and roman paganism is more absorbed into what some call Christianity than the other way around.
True followers of Christ abstain from all the pagan beliefs, traditions and celebrations that have been intertwined into the system.

1Co 10:21 You cannot drink the cup of the Lord and the cup of demons. You cannot partake of the table of the Lord and the table of demons.

Answer 2

The Byzantine Empire, originally the Eastern Roman Empire became Christianized when Constantine the Great made Christianity the state religion and then changed the name of his capital from Byzantium to Constantinople .. since his ancestry was Greek and his religion was, by choice, Christian, I think your question answers itself