Is there a way to prove the God of the Bible?

No, but all other existing theories have holes in them. Huge holes.

The current scientific model says it all started with the Big Bang and before that there was nothing. Yet, if an "explosion" creates a New York City, let alone occurs at all, then it is no longer nothing, but something, and no longer a valid answer to explain why we exist.

We can not say the Universe, the laws of physics, reason itself came from nothing. That is no answer.

You also can not say we came from an infinite series of Big Bangs. Think about it like this. Imagine I hand you a letter. You read it, then ask, "who wrote this?" Who created the letter, who wrote it?

I say, I got it from the postman (by our definition postmen do not write letters). You say, "No, who wrote it?" I say it came from an infinite series of postmen. That does not answer your question. We can not say we came from an infinite series of zero explanations.

There are other answers like, from the Great Eagle, things like that. But who made all the eagles?

Jesus claimed to be the God of the the Bible a few times, and he even used the scriptures to support his case (here is only a few verses to make this case):

Jesus claimed to be God:

Philip said to him, “Lord, show us the Father, and it is enough for us.” Jesus said to him, “Have I been with you so long, and you still do not know me, Philip? Whoever has seen me has seen the Father. How can you say, ‘Show us the Father’? (John 14:8-9 ESV)

I and the Father are one.” (John 10:30 ESV)

Jesus invoked the name of God that was revealed to Moses in Exd 3:14:

Your father Abraham rejoiced that he would see my day. He saw it and was glad.” So the Jews said to him, “You are not yet fifty years old, and have you seen Abraham?” Jesus said to them, “Truly, truly, I say to you, before Abraham was, I am.” So they picked up stones to throw at him, but Jesus hid himself and went out of the temple. (John 8:56-59 ESV)

Jesus claimed to fulfil a reading from Isaiah (61:1-2a):

And the scroll of the prophet Isaiah was given to him. He unrolled the scroll and found the place where it was written, “The Spirit of the Lord is upon me, because he has anointed me to proclaim good news to the poor. He has sent me to proclaim liberty to the captives and recovering of sight to the blind, to set at liberty those who are oppressed, to proclaim the year of the Lord's favor.” And he rolled up the scroll and gave it back to the attendant and sat down. And the eyes of all in the synagogue were fixed on him. And he began to say to them, “Today this Scripture has been fulfilled in your hearing.” (Luke 4:17-21 ESV)

Jesus used the scriptures to explaine his death and resurrection:

Was it not necessary that the Christ should suffer these things and enter into his glory?” And beginning with Moses and all the Prophets, he interpreted to them in all the Scriptures the things concerning himself. (Luke 24:26-27 ESV)

Paul also held that Jesus' death and resurrection was from the scriptures:

Paul, a servant of Christ Jesus, called to be an apostle, set apart for the gospel of God, which he promised beforehand through his prophets in the holy Scriptures, concerning his Son, who was descended from David according to the flesh and was declared to be the Son of God in power according to the Spirit of holiness by his resurrection from the dead, Jesus Christ our Lord, (Romans 1:1-4 ESV)

And Paul stated that if Jesus did not rise from the dead (according to the scriptures) then the Christian faith is futile and Christians are misrepresenting God:

For I delivered to you as of first importance what I also received: that Christ died for our sins in accordance with the Scriptures, that he was buried, that he was raised on the third day in accordance with the Scriptures, (1 Corinthians 15:3-4 ESV)

And if Christ has not been raised, then our preaching is in vain and your faith is in vain. We are even found to be misrepresenting God, because we testified about God that he raised Christ, whom he did not raise if it is true that the dead are not raised. (1 Corinthians 15:14-15 ESV)

Jesus and Paul seemed to think that the resurrection of Jesus proved that Jesus was the God of the Bible. If you can prove the resurrection then you have a good case that Jesus is who he said he was. There is no way to prove this scientifically, but historically there is some evidence that may imply it happened. This also depends on how strongly you take the historical sources, but most historians can agree that:

1. Jesus died by crucifixion.
2. Jesus was honorably buried in an easily accessible public tomb. 
3. The death of Christ caused His followers to lose all hope in His Messianic claims
4. Three days later, the tomb was empty.
5. The disciples had genuine experiences that they were convinced were literal appearances of the risen Christ. 
6. The disciples were radically transformed from skeptics and doubters to bold proclaimers of Christ’s Resurrection. 
7. Eleven of the twelve apostles suffered martyrs’ deaths for their convictions.
8. The Resurrection message was absolutely central to the early preaching of the Church.
9. The Resurrection message was central to the entire New Testament. 
10. The Resurrection was first proclaimed in the very environment most hostile to it, Jerusalem. Even there, those motivated to disprove the Resurrection could not do so. 
11. The Church exists only because of the disciples’ conviction that the Resurrection occurred.
12. The Sabbath Day was changed to Sunday.
13. James, Paul, and many other skeptics were only convinced as a result of personally seeing the risen Jesus

Some other resurrection theories (swoon, stolen body, hallucination, mistaken identity, wrong tomb etc...) try and account for either (1) the empty tomb, (2) the resurrection appearances and (3) the transformed lives of his followers, but no theory, besides Jesus' resurrection adequately explains all three.

Source

Resurrect? An Atheist and Theist Dialogue, Gary R. Habermas and Antony G. N. Flew, edited by John F. Ankerberg

No, there are no scientific experiments that you could run to show the existence of God.

When Christians talk about proving God, they mean something along the lines of Malachi 3:10 (NRSV)

Bring the full tithe into the storehouse, so that there may be food in my house, and thus put me to the test, says the Lord of hosts; see if I will not open the windows of heaven for you and pour down for you an overflowing blessing.

Another oft quoted verse is James 1:5-8 (NRSV)

If any of you is lacking in wisdom, ask God, who gives to all generously and ungrudgingly, and it will be given you. But ask in faith, never doubting, for the one who doubts is like a wave of the sea, driven and tossed by the wind; for the doubter, being double-minded and unstable in every way, must not expect to receive anything from the Lord.

In other words, to prove (test) God.

1. Believe in His word.
2. Ask God for the things he's already promised us.
3. Ask in sincere faith.
4. See God's blessings in your life.

First, it depends what you mean by proof.

1. Mathematical proof
2. Scientific proof
3. Court of law proof

The first of these is the strongest: you can build absolute proofs that cannot be refuted by anyone who accepts the principles of Mathematics. However, such proofs are limited to the perfect abstractions described by mathematicians, and don't necessarily work that well in the real world. For example, you can prove that 1 + 1 = 2, but you can't easily prove that 1 haystack + 1 haystack = 2 haystacks because haystack counting doesn't follow the normal rules (1 haystack + 1 haystack = 1 big haystack). As a result, you can't apply that kind of proof to God.

The second is weaker, but is still strong: scientific proof is the result of trying to find evidence against a theory, and accepting it as proven only if no evidence has yet been found that undermines that theory. If such evidence is found, and if the theory can't be adjusted to accommodate it, then the theory must be rejected. Nevertheless, if such evidence can't be found by the scientific community then there is good reason to accept a theory as proven. Some people feel that this kind of proof does apply to God, but there isn't sufficient consensus to accept that God's existence has been proven in this way.

Finally, the evidence required to convict someone in a court of law is somewhat weaker. The community of investigators is far smaller than that required for scientific proof, and the process of proof is far less rigorous than scientific proof would be. Whereas scientific proof relies on repeatability, this is generally impossible when someone is on trial. Now, despite the relative weakness of courtroom proof, it is often the best we've got, and we have to make real-world (even life-and-death) decisions based on this kind of proof. Many of us have looked the evidence for and against the Christian God's existence and concluded that, to the best of our ability, we have found sufficient 'proof' that He is real. We believers are willing to make real-world decisions on the basis of this proof, but recognize that it is at best a court-room proof rather than a mathematical or even a scientific proof.

For the most part, we believers assume that God's existence is logical. Some of us are able to articulate our understanding of that logic, whilst others are willing to accept that such a logic may exist but that we either don't or even can't know what it is.

As to what evidence people use as the basis of their proof, there are numerous sources, and different believers find different kinds of evidence persuasive. Some popular sources of evidence include:

1. Religious experience - either an "inner witness" or observable phenomena attributed to God or to the results of faithful obedience.
2. The testimony of other believers
3. Historical records, including (but not limited to) the Bible
4. Philosophical arguments - such as Pascal's Wager, Lewis' Trilemma, Aquinas arguments (as paraphrased on David Laberge's answer), etc. This includes both arguments for God's existence and refutations of arguments against God's existence.
5. Let's be honest - wishful thinking and guesswork

The bottom line is this: we each of us have the option to look at the evidence available for God's existence and makes up their own mind. There is no absolute certainty - but for us Christians, there is enough.

In Mere Christianity, CS Lewis describes such a proof (source):

My argument against God was that the universe seemed so cruel and unjust. But how had I got this idea of just and unjust? A man does not call a line crooked unless he has some idea of a straight line. What was I comparing this universe with when I called it unjust? If the whole show was bad and senseless from A to Z, so to speak, why did I, who was supposed to be part of the show, find myself in such violent reaction against it? A man feels wet when he falls into water, because man is not a water animal: a fish would not feel wet.

Of course I could have given up my idea of justice by saying it was nothing but a private idea of my own. But if I did that, then my argument against God collapsed too- for the argument depended on saying that the world was really unjust, not simply that it did not happen to please my private fancies. Thus in the very act of trying to prove that God did not exist-in other words, that the whole of reality was senseless-I found I was forced to assume that one part of reality-namely my idea of justice-was full of sense.

Consequently atheism turns out to be too simple. If the whole universe has no meaning, we should never have found out that it has no meaning: just as, if there were no light in the universe and therefore no creatures with eyes, we should never know it was dark. Dark would be without meaning.

In other words, we would not have even a concept of justice, if absolute justice did not exist, just like a fish has no concept of either dry or wet.

CS Lewis also had his famous trilemma, demonstrating that if you accept the existence of Jesus and what He said, He must have been God. Nicky Gumbel (I think) summarised it as Jesus can only have been Mad, Bad or God."

Here's the trilemma in full:

"I am trying here to prevent anyone saying the really foolish thing that people often say about Him: I’m ready to accept Jesus as a great moral teacher, but I don’t accept his claim to be God. That is the one thing we must not say. A man who was merely a man and said the sort of things Jesus said would not be a great moral teacher. He would either be a lunatic — on the level with the man who says he is a poached egg — or else he would be the Devil of Hell. You must make your choice. Either this man was, and is, the Son of God, or else a madman or something worse. You can shut him up for a fool, you can spit at him and kill him as a demon or you can fall at his feet and call him Lord and God, but let us not come with any patronising nonsense about his being a great human teacher. He has not left that open to us. He did not intend to. ... Now it seems to me obvious that He was neither a lunatic nor a fiend: and consequently, however strange or terrifying or unlikely it may seem, I have to accept the view that He was and is God.

The classic arguments I know are the following :

1. Cosmological argument: That every thing that has a beginning must have a cause.
2. Theological argument: The order and the useful setting in a system imply an intelligence and a finality that did the organization.
3. Ontological argument: That the idea of God itself is a proof for the existence of God. How can a finite being imagine an infinite one?
4. Moral argument: The conscience of the human is proof of a conscience giver. Or, the morality within man means that there must be a moral law giver.
5. Congruence argument: This argument maintains that the previous arguments present the best way to understand the person of God. That the belief in the existence of God is the better way to explain our moral nature (mental and religious). Also, it is the better way to explain the material universe.

These arguments work together to form one argument. So please don't work with these as 5 single, independent arguments.