One of the little oddities of Christianity is that while the divinity of Jesus is arguably the defining feature of our faith, it's surprisingly difficult to find this doctrine explicitly taught in Scripture. Sure, there are a few verses that apply the name "God" to Jesus (such as John 1:1), but those are few and far between, and they're often grammatically ambiguous. Instead, if we want to build a rock-solid biblical case for the divinity of Jesus, we have to look at other, much subtler clues.
God and Israel
Those subtle clues can be hard to find, but once we know what to look for, we can see them all over the Bible. For example, there's an apparent discrepancy between the Old and New Testaments that can only be resolved if Jesus really is God. In the Old Testament, God promised his people that he would one day marry them:
You shall no more be termed Forsaken,
and your land shall no more be termed Desolate;
but you shall be called My delight is in her,
and your land Married;
for the Lord delights in you,
and your land shall be married.
For as a young man marries a virgin,
so shall your sons marry you,
and as the bridegroom rejoices over the bride,
so shall your God rejoice over you. (Isaiah 62:4-5)
And in that day, says the Lord, you will call me, "My husband," and no longer will you call me, "My Baal"...And I will betroth you to me forever; I will betroth you to me in righteousness and in justice, in steadfast love, and in mercy. I will betroth you to me in faithfulness; and you shall know the Lord. (Hosea 16:19-20)
In both of these passages, God promised to redeem his people Israel, and he used the metaphor of marriage to describe their future redemption. He said that when he saves them, he will be their husband, and they will be his bride.
Jesus is the Bridegroom
That seems simple enough, but when we turn to the New Testament, things get a bit complicated. When this promised salvation finally comes in Jesus Christ, we find that God's people are married to him, not God. For example, we find some hints of this in the Gospels:
John [the Baptist] answered, "No one can receive anything except what is given him from heaven. You yourselves bear me witness, that I said, I am not the Christ, but I have been sent before him. He who has the bride is the bridegroom; the friend of the bridegroom, who stands and hears him, rejoices greatly at the bridegroom's voice; therefore this joy of mine is now full. He must increase, but I must decrease." (John 3:27-30)
Then the disciples of John came to him, saying, "Why do we and the Pharisees fast, but your disciples do not fast?" And Jesus said to them, "Can the wedding guests mourn as long as the bridegroom is with them? The days will come, when the bridegroom is taken away from them, and then they will fast." (Matthew 9:14-15)
The Church is the Bride
Both of these passages call Jesus "the bridegroom," but they don't tell us who his bride is. For that, we need to turn to one of the letters of St. Paul:
"For this reason a man shall leave his father and mother and be joined to his wife, and the two shall become one flesh." This is a great mystery, and I mean in reference to Christ and the church. (Ephesians 5:31-32)
In this text, Paul begins by quoting a verse about marriage from the creation stories in Genesis (Genesis 2:24), and then he tells us that he's applying it "to Christ and the Church." He's saying that Jesus' bride is the Church, complementing the "bridegroom" passages in the Gospels and giving us the information they omit.
And in case there's any doubt left, let's look at the last book of the Bible, Revelation:
And I saw the holy city, new Jerusalem, coming down out of heaven from God, prepared as a bride adorned for her husband; and I heard a great voice from the throne saying, "Behold, the dwelling of God is with men. He will dwell with them, and they shall be his people, and God himself will be with them"...Then came one of the seven angels who had the seven bowls full of the seven last plagues, and spoke to me, saying, "Come, I will show you the Bride, the wife of the Lamb." And in the Spirit he carried me away to a great, high mountain, and showed me the holy city Jerusalem coming down out of heaven from God. (Revelation 21:2-3, 9-10)
In this passage, which describes the end of time and the consummation of human history, we see very clearly that the New Jerusalem, a symbol of the people of God, is married to "the Lamb." And who is this lamb? Throughout the book of Revelation, it's Jesus. The book calls him this because he was sacrificed for our sins (for example, Revelation 5:6, 12; 12:11), much like the Israelites in the Old Testament sacrificed animals to atone for their sins. More specifically, they sacrificed lambs during the feast of Passover, a feast that celebrated their rescue from Egypt, and that redemption foreshadowed the greater salvation that Jesus would later win for us (1 Corinthians 5:7).
Jesus, Our Divine Bridegroom
From all this, we can see that the husband of God's people is Jesus, and he will remain so forever and ever. Now, if we don't believe that Jesus is God, we are forced to conclude that God's promise to marry his people will never actually be fulfilled; instead, we have to settle for Jesus.
However, for people who believe that the Bible is the word of God, that's unacceptable. God always fulfills his promises, so there has to be another way of looking at this apparent discrepancy. And that other way is to conclude that Jesus really is God. God promised to marry his people, and he makes good on that promise in the person of Jesus Christ, who is himself God.
Like I said earlier, this is a subtle indication of Jesus' divinity. We can easily overlook it if we don't know it's there, but once we see it, the doctrine becomes impossible to deny.