Strictly from the gospel accounts, the expectation of the Jewish Messiah comes through clearly. For example, Simeon and Anna (Luke 2) both understood the importance of Jesus's birth, and Simeon in particular detailed the Messiah's role as "a light of revelation to the Gentiles [non-Jewish nations]." When the magi arrived (Matthew 2), the scholars in Israel directed them to Bethlehem as the Messiah's birthplace.
John the Baptist, in answer to the priests and Levites sent from Jerusalem, immediately confesses he is not the Messiah (John 1:20). The apostle Andrew calls his brother, Peter, to Jesus by saying, "We have found the Messiah" (John 1:41). Later, the Samaritan woman at the well knows the Messiah will come (John 4:25), and the crowds listening to Jesus argued not about the reality of a Messiah, but His place of origin and what signs were appropriate (John 7:27-31).
In Acts 5:36-37, Gamaliel hints at the Messiah "fever" of the age. Two other men at around the time of Jesus had gathered a following by claiming to be the Christ. Neither, however, could fulfill the prophetic requirements, and consequently their followers dispersed.
Beyond the Bible, Jewish rabbis had long expected and made reference to the Messiah based mostly on Old Testament prophecies. A review shows that their statements align with the New Testament fulfillment: His existence before the creation of the world; His preeminence over Moses and the angels; His sufferings; His violent death for His people; His kingdom; and others. However, their expectations also included speculation beyond Scripture, which is why many rejected Jesus as Messiah.
In addition, several works written prior to, concurrent with, and soon after Jesus's life and ministry make reference to a coming Messiah. These works, called the Pseudepigrapha because the authors wrote under false names, offer glimpses into the expectations of the Jewish people scattered throughout the Roman Empire and beyond.
Adapted from The Life and Times of Jesus the Messiah by Alfred Edersheim (Book II, Chapter V) with additions from the lecture notes of Dr. Doug Bookman, professor of New Testament Exposition at Shepherds Theological Seminary (used by permission).