John summarizes the purpose of his letter in 20:30-31. “These are written that you might believe.” If we follow the aorist subjunctive reading of the Greek, John wants to bring people to faith in Jesus Christ the Son of God. If we follow the present tense it might be that he wants his readers to continue to believe (the NEB has “keep the faith”). In either case if John's letter can keep people in the faith it can bring them to faith to begin with. If he's writing to believers maybe their staying in the faith is his prime concern. But again, we've all known people who have come to faith in Jesus Christ through the reading of the book of John so it hardly makes sense to ignore that on the basis of a variant reading of a single word. Though I tend to think he is establishing people in the true faith, to deny the evangelistic nature of the book seems to be over the top.
My impression is that John wants to establish the humanity, divinity and supremacy of Jesus of Nazareth, the Messiah. He says he wants to show that Jesus is “the Christ,” the one who fulfils all that the Jewish faith looked for. In doing this he shows that Christ is greater than the Baptist, greater than Moses, greater than Jacob and greater that Abraham. He wants to show that Jesus (of Nazareth ) the Messiah was also “the Son of God”. It's clear enough that “son of God” can and does act as a Messianic title, a title of royalty that others share, but I'm sure there's more to it than that. With Jesus, “sonship” has another dimension (see John 5:18 and 10:33 ) that is unique to him. Jesus of Nazareth was not just a man he was God being a man (1:1,14).
I get the impression that John wants to establish believers in the true faith that included the humanity, divinity and fullness of person and vocation of the one they had come to believe in. This suggests that there was a need for such a development of the gospel and maybe that means there was a growing Gnosticism in Asia Minor . So contrary to one strand of Gnostic leanings John insists that Christ was indeed truly human. Contrary to the opposite Gnostic tendency John insists that Christ is truly God and contrary to any Jewish sectarian views Christ is greater than the Baptist or any other Jewish figure.
All this he wants to establish but not so that they can have correct views--end of story. He wants them to have life (see John 17:3)! The true faith is the medium by which life with God comes. It would be silly to say that what we believe doesn't matter because that would be a flat contradiction not only to what John records (see 8:32, for example) but also to his stated purpose. Still, life doesn't come because we can give correct answers to a religious quiz. It comes from God through Jesus Christ to those who embrace with glad trusting commitment the truth about God brought to us in and through Jesus Christ the Son of God.