Details of Buddha's birth come from Buddhist literature. It's hard to say how much is historical and how much just tradition.

Tradition does indeed say he was born in a Hindu family. His father was Sudhodhan, a minor Kshatriya king or chieftain of the Shakya clan. Their kingdom was called Kosala, with its capital at Kapilvastu. Kshatriyas are the Hindu warrior caste.

His mother was Maya Devi of the Koliya clan, also Kshatriya by caste. The Koliyas and Shakyas traditionally intermarried only from each other's clan, and had done so for generations before Buddha. Maya Devi and her sister Mahapajapati were Koliyas, and both were married to Sudhodhan of the Shakya clan. Maya Devi died soon after giving birth to Buddha, so he was raised by his aunt/stepmother Mahapajapati. In turn, when Buddha was older, he married Yashodhara who was also a princess of the Koliya clan and his first cousin.

The Hinduism of his time was not identical to Hinduism today. Of course, Hinduism today isn't monolithic either, it consists of a wide range of beliefs. In Buddha's time, it was more ritualistic and closer to the Vedas. The Hinduism of his time is often referred to as "Brahmanism" in western literature, though in India it has no special name, it's just the old Sanatan Dharma, the Hindu orthodoxy. It's not quite Vedic Hinduism - the Vedas were already ancient in Buddha's time - but rather the Vedic gods combined with the philosophical influence of the Upnishads, the oldest of which had been written a few hundred years before Buddha's birth.

This was a time of some competition between brahmanic and shramanic traditions. The Brahmans were the orthodoxy, who stuck to rigid Vedic rites. The shramanic tradition was more monk-like - semi-ascetic groups of people in search of enlightenment, who often lived in forest communes. They were not Brahmans by caste and did not generally follow the caste system.

Buddha's preaching is full of criticism for the Brahmans and their elaborate ceremonies and Vedic rituals and sacrifices, so we know he didn't think much of the Brahmans. He seems to have adopted some of the ideas of the shramanic traditions of his time, so that is perhaps the stronger Hindu influence on his thought than Brahmanism.