How do you give someone who doesn’t believe in the Bible any “proof” that Jesus actually said the things He is created to have said?

There IS a level of proof from outside the Bible that Christianity is founded on facts. One reason for this is the simple fact that Jesus Christ existed as a real person in real time. He invaded space and time in a real world. The following conclusions can be drawn from the ancient, non-Christian sources that we have:
(1) Jesus was a Jewish teacher.
(2) Many people believed he performed healings and exorcisms.
(3) Some people believed he was the promised Messiah (Christ) of prophecy.
(4) He was rejected by the Jewish religious leadership.
(5) He was crucified under the governor Pontius Pilate during the reign of the emperor Tiberius.
(6) Despite this shameful (criminal) death, his followers, who believed he was still alive, had spread beyond Palestine so much so that there were multitudes of them in Rome by 64 AD.
(7) All kinds of people from the cities and the countryside—men and women, slave and free—worshiped him as God.
In addition, there are two other main sources of information that allow us to trust the scriptures we have:
(1) Archaeology has literally completely proven the historical accounts found in the Biblical records, both Old and New Testaments. Luke in particular has been proven by archaeology to have been a meticulously detailed historian.
(2) All the accounts of Christ that we have in the New Testament were written within the first generation of his lifetime. The gospel of Mark may have been written within 30 years of Christ’s life on earth (late 50s to about 60 AD); Luke (and possibly Matthew) shortly thereafter (because some of Mark’s material is found word-for-word in those two gospels). Only John’s gospel was likely written significantly later—and still in the first century (likely the 80s AD). Where that is huge is the fact that direct eyewitnesses to the life of Christ would have still been alive when the gospels appeared; you would have had people quick to pounce if the gospel accounts were in error because they were there.
A great resource regarding this question is the book, The Historical Jesus by Gary Habermas. He looks at several levels of ancient documentation, including archaeology, ancient Christian writers and ancient non-Christian writers. Among his finds regarding ancient “non-Christian” sources:
(1) Tacitus, a Roman historian of the first and second century, who life covered several Roman emperors. He is known by scholars as a man of “integrity and essential goodness.” Tacitus tells us in his writings that a group called Christians existed, and that they named for their founder—Christus (the Latin form of Christ). This “Christus” was put to death by the Roman procurator (governor) Pontius Pilate during the reign of Tiberius (who reigned as emperor from 14-37 AD). His death was thought to end that particular belief for a short time, but it broke out again in Judea, the place of its origin. His followers carried his doctrine to Rome. The emperor Nero blamed Christians when a great fire broke out in Rome, and many Christians were arrested, not so much for being blamed for the fire but, in Tacitus’ words, “as of hatred against mankind.” They were subsequently tortured, including crucifixion and also being burned to death. Tacitus reported that the people of Rome actually began to feel compassion for the Christians as they suffered. It is thought by some scholars that Tacitus’ report that belief in Jesus, initially silenced, “again broke out” after a time in Judea could be an indirect reference to the resurrection which, according to the Scriptures, emboldened all of Christ’s followers as never before.
(2) Suetonius was also a Roman historian, and a secretary during the reign of Emperor Hadrian (117-138 AD). As such, he would have had access to official records. Writing about an incident during the reign of Emperor Claudius (41-54 AD), he notes that, “Because the Jews at Rome caused continuous disturbances at the instigation of ‘Chrestus,’ he expelled them from the city.” There is actually interesting Biblical corroboration for this incident; Luke writes in the book of Acts: After this (Paul speaking on Mars Hill in Athens), Paul left Athens and went to Corinth. There he met a Jew named Aquila, a native of Pontus, who had recently come from Italy with his wife Priscilla, because Claudius had ordered all the Jews to leave Rome. Paul went to see them—Acts 18:1-2. Suetonius also mentions the tortures Christians endured at the hand of Nero following the great fire; Suetonius refers to Christians as “a sect professing a new and mischievous religious belief.” The mischief was found in Christians’ assertion that Jesus, particularly by means of His resurrection, was divine, which obviously set them against the prevailing emperor worship of the time. Refusal to acknowledge the divinity of the emperor could make one guilty of treason, and therefore liable to some of the worst punishments of the time. Both Tacitus and Suetonius directly credit Christian beliefs as coming from Christ.
(3) Josephus, a Jewish historian of the first century, ultimately became the court historian for the emperor Vespasian. In his writings appears two references to Christ: the first a brief reference to James “the brother of Jesus who was called Christ.” This is the James who became the leader of the Jerusalem church and later wrote the epistle that bears his name. The second one is a more detailed writing, where Josephus is quoted as saying: “Now there was about this time Jesus, a wise man, if it be lawful to call him a man. For he was one who wrought surprising feats . . . He was (the) Christ . . . he appeared to them alive again the third day, as the divine prophets had foretold these and ten thousand other wonderful things concerning him.” Early church fathers were divided on this passage; Origen says that Josephus never believed Jesus to be the Messiah, while Eusebius includes the whole quote in his writings. Oddly enough, an Arabic manuscript from Josephus includes this account, and more: “At this time there was a wise man who was called Jesus. His conduct was good and (he) was known to be virtuous. And many people from among the Jews and the other nations became his disciples. Pilate condemned him to be crucified and to die. But those who had become his disciples did not abandon his discipleship. They reported that he had appeared to them three days after his crucifixion, and that he was alive; accordingly he was perhaps the Messiah, concerning whom the prophets had recounted wonders.”
(4) Pliny the Younger was the governor of Bithynia, an area of Asia Minor (modern day Turkey). He left letters to the government in Rome as to how to deal with the Christians. Because of them and their influence, he discovered that pagan temples were nearly deserted, pagan holidays hardly observed at all, and the animals raised for sacrifice had few buyers. In investigating them, he wrote, “They (the Christians) were in the habit of meeting on a certain fixed day before it was light, when they sang in alternate verses a hymn to Christ, as to a god, and bound themselves by a solemn oath, not to any wicked deeds, but not to commit any fraud, theft or adultery, never to falsify their word, nor deny a trust when they should be called upon to deliver it up; after which it was their custom to separate, and then reassemble to partake of food—but food of an ordinary and innocent kind.”
(5) The Talmud was a collection of Jewish oral tradition handed down from generation to generation, written down and ordered according to subject, finally completed about 200 AD. The actual writing was called the Mishnah. A commentary on these writings was also completed, called the Gemaras. The two collections together make up the Talmud. One particular writing is very interesting: “On the eve of the Passover Yeshua was hanged. For forty days before the execution took place, a herald went forth and cried, ‘He is going forth to be stoned because he has practiced sorcery and enticed Israel to apostasy. Anyone who can say anything in his favor, let him come forward and plead on his behalf.’ But since nothing was brought forward in his favor he was hanged on the eve of the Passover.” Two things to note: First, the reference to hanging is actually supported by Scripture. According to the apostle Paul in his letter to the Galatians, Christ redeemed us from the curse of the law by becoming a curse for us, for it is written: “CURSED IS EVERYONE WHO IS HUNG ON A TREE”—Paul, Galatians 3:13 (quoting Deuteronomy 21:23). Also, regarding the reference to Jesus being stoned, it was of course the Biblical sanction; God Himself spoke these words in the book of Leviticus: Say to the Israelites: ‘If anyone curses his God, he will be held responsible; anyone who blasphemes the name of the Lord must be put to death. The entire assembly must stone them. Whether an alien or native-born, when he blasphemes the Name, he must be put to death—God speaking, Leviticus 24:15-16. And while that did not happen the case of Christ, Scripture certainly records Jesus being threatened with this; the gospel of John specifically tells us, Your father Abraham rejoiced at the thought of seeing my day; he saw it and was glad.” “You are not yet fifty years old,” the Jews said to him, “and you have seen Abraham!” “I tell you the truth,” Jesus answered, “before Abraham was born, I am!” At this, they picked up stones to stone him, but Jesus hid himself, slipping away from the temple grounds—John 8:56-59. A couple of chapters later, John records this: I and the Father are one.” Again the Jews picked up stones to stone him, but Jesus said to them, “I have shown you many great miracles from the Father. For which of these do you stone me?” “We are not stoning you for any of these,” replied the Jews, “but for blasphemy, because you, a mere man, claim to be God”—John 10:30-33. That being said, this is excellent attestation of non-Christian support for the events we find in the Bible, and supporting the idea that Jesus taught unlike anyone else, that he had disciples and that, following his death and resurrection, the disciples went out boldly to preach his gospel to everyone. Within a few short years in historical time, a “sect” operating in distant Palestine was rattling Rome and its government with this new teaching, over which they would never be able to exert control. An idea defeated an empire. These are just some of the ancient sources that we find, both Christian and non-Christian, that support the New Testament scriptures.
Beyond this, however, we must remember that, as we reach out to others with the gospel, we are not working alone—the Holy Spirit accompanies us and guides us in all these encounters. And of course, we also must remember that unfortunately, for some, no matter the argument or the proofs, they will not believe. They will not accept the Scriptures. They do not want to “bow the knee” . . . to anybody. And as Christ Himself pointed out, If they do not listen to Moses and the Prophets, they will not be convinced even if someone rises from the dead—Jesus Christ, Luke 16:31.