When Jesus clears the temple of the money changers:

John 2:17 His disciples remembered that it was written, “Zeal for your house will consume me.”

The Jews ask for a sign:

John 2:18 So the Jews said to him, “What sign do you show us for doing these things?” 19 Jesus answered them, “Destroy this temple, and in three days I will raise it up.” 20 The Jews then said, “It has taken forty-six years to build this temple, and will you raise it up in three days?” 21 But he was speaking about the temple of his body. 22 When therefore he was raised from the dead, his disciples remembered that he had said this, and they believed the Scripture and the word that Jesus had spoken.

John says that Jesus is a witness of God:

John 3:31 He who comes from above is above all. He who is of the earth belongs to the earth and speaks in an earthly way. He who comes from heaven is above all. 32 He bears witness to what he has seen and heard, yet no one receives his testimony. 33 Whoever receives his testimony sets his seal to this, that God is true. 34 For he whom God has sent utters the words of God, for he gives the Spirit without measure. 35 The Father loves the Son and has given all things into his hand. 36 Whoever believes in the Son has eternal life; whoever does not obey the Son shall not see life, but the wrath of God remains on him.

The Samaritan woman testifies to what Jesus said to her (he told me all that I ever did) but many believed because of his word (4:39-42).
Jesus’ second sign causes belief in official when Jesus healed his son from a distance (John 4:53-54).
Jesus’ third sign was to heal a man on the Sabbath who had been an invalid for thirty-eight years (John 5:1-9).
The man who had been healed by Jesus testified that he had been healed (John 5:13-15).
Jesus tells the Pharisees of all who testify of him:

John 5:30 “I can do nothing on my own. As I hear, I judge, and my judgment is just, because I seek not my own will but the will of him who sent me. 31 If I alone bear witness about myself, my testimony is not deemed true. 32 There is another who bears witness about me, and I know that the testimony that he bears about me is true. 33 You sent to John, and he has borne witness to the truth. 34 Not that the testimony that I receive is from man, but I say these things so that you may be saved. 35 He was a burning and shining lamp, and you were willing to rejoice for a while in his light. 36 But the testimony that I have is greater than that of John. For the works that the Father has given me to accomplish, the very works that I am doing, bear witness about me that the Father has sent me. 37 And the Father who sent me has himself borne witness about me. His voice you have never heard, his form you have never seen, 38 and you do not have his word abiding in you, for you do not believe the one whom he has sent. 39 You search the Scriptures because you think that in them you have eternal life; and it is they that bear witness about me, 40 yet you refuse to come to me that you may have life. 41 I do not receive glory from people. 42 But I know that you do not have the love of God within you. 43 I have come in my Father’s name, and you do not receive me. If another comes in his own name, you will receive him. 44 How can you believe, when you receive glory from one another and do not seek the glory that comes from the only God? 45 Do not think that I will accuse you to the Father. There is one who accuses you: Moses, on whom you have set your hope. 46 If you believed Moses, you would believe me; for he wrote of me. 47 But if you do not believe his writings, how will you believe my words?”

Jesus’ fourth sign is the feeding of the 5,000 (John 6:1-15) and when the people see the sign they believe that Jesus is the Prophet (John 6:14).
Jesus’ fifth sign is that he walks on water to the disciples (John 6:16-21).
The people wanted to make Jesus king and they sought him out not because of the signs but because of the bread (John 6:26-27).
After this they ask Jesus for a sign, Moses gave them bread what will He give them:

John 6:28 Then they said to him, “What must we do, to be doing the works of God?” 29 Jesus answered them, “This is the work of God, that you believe in him whom he has sent.” 30 So they said to him, “Then what sign do you do, that we may see and believe you? What work do you perform? 31 Our fathers ate the manna in the wilderness; as it is written, ‘He gave them bread from heaven to eat.'” 32 Jesus then said to them, “Truly, truly, I say to you, it was not Moses who gave you the bread from heaven, but my Father gives you the true bread from heaven. 33 For the bread of God is he who comes down from heaven and gives life to the world.” 34 They said to him, “Sir, give us this bread always.” 35 Jesus said to them, “I am the bread of life; whoever comes to me shall not hunger, and whoever believes in me shall never thirst. 36 But I said to you that you have seen me and yet do not believe. 37 All that the Father gives me will come to me, and whoever comes to me I will never cast out. 38 For I have come down from heaven, not to do my own will but the will of him who sent me. 39 And this is the will of him who sent me, that I should lose nothing of all that he has given me, but raise it up on the last day. 40 For this is the will of my Father, that everyone who looks on the Son and believes in him should have eternal life, and I will raise him up on the last day.”

Jesus’ sixth sign was to heal a man on the Sabbath who had been blind from birth (John 9:1-7).
The man healed of blindness testifies of Jesus and rebukes the Pharisees for their unbelief:

John 9:30-34The man answered, “Why, this is an amazing thing! You do not know where he comes from, and yet he opened my eyes. 31 We know that God does not listen to sinners, but if anyone is a worshiper of God and does his will, God listens to him. 32 Never since the world began has it been heard that anyone opened the eyes of a man born blind. 33 If this man were not from God, he could do nothing.” 34 They answered him, “You were born in utter sin, and would you teach us?” And they cast him out.

Jesus tells the blind man that he healed that he has seen the Son of God and the man believed him and worshiped him (John 9:35-38).
The Pharisees ask Jesus if he is the Christ (John 10:24) and Jesus replies that the works testify of him:

John 10:24 So the Jews gathered around him and said to him, “How long will you keep us in suspense? If you are the Christ, tell us plainly.” 25 Jesus answered them, “I told you, and you do not believe. The works that I do in my Father’s name bear witness about me, 26 but you do not believe because you are not part of my flock.

Jesus tells them that he and the Father are one and they pick up rocks to stone him since they believe he blasphemed God (John 10:31-33).
Jesus tells them that they should believe because of the works that he has done (John 10:37-38). And their response was to try to arrest him (John 10:39). Though some of the Jews wanted to stone him, others believed (John 10:40-42):
Jesus seventh sign is the raising of Lazarus from the dead:

John 11:4 But when Jesus heard it he said, “This illness does not lead to death. It is for the glory of God, so that the Son of God may be glorified through it…14 Then Jesus told them plainly, “Lazarus has died, 15 and for your sake I am glad that I was not there, so that you may believe. But let us go to him…” 25 Jesus said to her, “I am the resurrection and the life. Whoever believes in me, though he die, yet shall he live, 26 and everyone who lives and believes in me shall never die. Do you believe this?” 27 She said to him, “Yes, Lord; I believe that you are the Christ, the Son of God, who is coming into the world…” 40 Jesus said to her, “Did I not tell you that if you believed you would see the glory of God?” 41 So they took away the stone. And Jesus lifted up his eyes and said, “Father, I thank you that you have heard me. 42 I knew that you always hear me, but I said this on account of the people standing around, that they may believe that you sent me.

Notice the emphasis on the death in John 11:44 “The man who had died came out.”
Many believed having seen Lazarus raised from the dead (John 11:45).
The Pharisees see that Jesus performed many signs but they planned to kill him (John 11:46-53). They not only plotted to kill Jesus but Lazarus as well because many believed in Jesus because of him (John 12:10-11).
OT fulfillment of prophecy demonstrates to the disciples the truth of Christ’s claim to be King:

John 12:12-16 The next day the large crowd that had come to the feast heard that Jesus was coming to Jerusalem. 13 So they took branches of palm trees and went out to meet him, crying out, “Hosanna! Blessed is he who comes in the name of the Lord, even the King of Israel!” 14 And Jesus found a young donkey and sat on it, just as it is written, 15 “Fear not, daughter of Zion; behold, your king is coming, sitting on a donkey’s colt!” 16 His disciples did not understand these things at first, but when Jesus was glorified, then they remembered that these things had been written about him and had been done to him.

The crowd, who had been at the resurrection of Lazarus, bear witness to others (John 12:17-18).
God the Father is a witness from heaven that He hears His Son’s prayers (John 12:28-30).
Jesus says that they are to believe in the light:

John 12: 35 So Jesus said to them, “The light is among you for a little while longer. Walk while you have the light, lest darkness overtake you. The one who walks in the darkness does not know where he is going. 36 While you have the light, believe in the light, that you may become sons of light.” When Jesus had said these things, he departed and hid himself from them.

The people did not believe the signs (John 12:37) which fulfills Isaiah (John 12:37-41).
Some of the officials believed (John 12:42).
Christ came as light so that those who believe in him may not remain in darkness (John 12:44-46)
Jesus tells the disciples that one of them will betray him and that he is telling them so that they “may believe that I am he” (John 13:19).
The works are a witness to the disciples (John 14:10-11).
The Father is glorified by the fruit of the disciples (John 15:8).
Jesus and the Father are hated because of the works (John 15:24-25).
John makes sure the reader understands that Jesus said beforehand that Peter would deny him and that he told them the manner of his death (John 18:32).
Jesus said that he was born to bear witness to the truth (John 18:37).
John states after each event that happened during the crucifixion that it fulfills Scripture (John 19:24, 36-37).
John states that he bore witness to Jesus’ death on the cross (John 19:36).
John saw the linen cloths and believed (John 19:8-9).
Thomas believed when he say the resurrected Jesus and Jesus tells him that those who believe, even though they haven’t seen him, are blessed (John 20:24-29).
Jesus did many signs, these were written so that the reader would believe (John 21:30-31).
John notes that the disciples had seen Jesus three times since his death (John 21:14).
John bears witness to the truth of his report (John 21:24).
John clearly fulfills his stated purpose. We can see throughout the Gospel, Jesus performs sign after sign and the ultimate sign is his resurrection. All other signs pale in comparison to this sign. Each sign should have elicited belief because each sign was a something that clearly demonstrated a power beyond anything a human could do. When we read the Gospel of John we are faced with the same situation as the Jews, are we going to believe? John clearly makes his case, presenting clear evidence that he himself witnessed. He saw what happened and he testifies to it.
I’m assuming that if you are reading this, that you are a Christian but if you’re not, read the Gospel of John yourself and then see if he makes his case. And if you are a Christian, give this Gospel to your unbelieving friends and say, “Do me the favor of just reading this straight through and see if you were in a courtroom, would you accept the evidence of this eyewitness?”