Jesus demonstrated how the Scriptures and the miracles serve as evidence for His legitimacy. However, the Jewish leadership in John 5 did not even bother seeking knowledge on Jesus's legitimacy (unlike Nicodemus) and went straight to rejecting Jesus.
First off, Jesus talks about how he "do[es] not receive honor from men" (5:41). From the context as well as similar use in 5:34, Jesus means that getting honor is not his purpose, and the Greek word for "receive" (Strong 2983) that can also mean "strive to obtain" supports the meaning. Rather than coming for men's honor, Jesus came for the Father (5:42). However, Jesus noted that not only do the Jewish leadership reject Jesus, but based on their attitude, they are willing to accept other leaders that "comes in his own name" (5:43).
Additionally, Jesus talks about how believing in Jesus can come from believing in Moses' writings and seeking the honor of God (5:44, 46). Moses had foreshadowed a person now known as Jesus in Deuteronomy 18, and the signs help provide support for Jesus's legitimacy. However, because Moses foreshadowed Jesus and the Jewish leaders outright rejected Christ, Jesus knew that the peeps truly didn't care about God. They may be knowledgeable in the Mosaic Law, but they missed a crucial point.