What was the relationship between greek and roman culture ?

While both Greek and Romans were pretty ethnocentric by modern standards, the Romans assimilated far more people into their institutional lives.

Many non-Greeks adopted Gteek lifestyles, language and habits after the age of Alexander, but the cross-pollination was more frequently cultural than political. Cleopatra might have dressed like an Egyptian queen and patronized the Egyptian gods, but she wouldn't have had Egyptian generals or Egyptian judges. The Greeks tended to settle into the cultures they occupied like the British in India: remaining separate from and believing themselves superior to the people around them, even while encouraging the 'natives' to adopt their culture habits.

Romans did a much more thorough job assimilating the peoples they conquered. Non-Romans could and did become citizens, even from very early times. This started with neighboring groups like the Latins, but eventually extend to the rest of Italy and later to the whole empire. Eventually there would be "Roman" emperors of Syrian, British, Spanish, Gallic, Balkan, and North African descent Farther down the social scale the mixing was much more complete (enough to irritate many Roman traditionalists). This wasn’t just a practical accommodation, either — when emperor Claudius allowed Gauls into the Roman Senate he pointed out that by his time the Romans had been assimilating former enemies since the days of Aeneas.