For me, it boils down to these.
(1) For me, it is self-evident that the things in nature have a Creator.
I appreciate that atheists and skeptics do not see it this way, even though they will accept that everything else in life has a creator. In other words, they use, say, an Apple computer knowing it was created, yet they will eat an apple and assume it simply came about as a result of random chance over a huge period of time.
Actually, the apple WAS produced... by the tree. The tree was produced... by God. But who produced God? Skeptics and atheists have the same problem with the Universe. They believe the Universe had an origin, a "First Cause" if you will... yet what caused that?
So really, skeptics and believers have the same problem in relation to the First Cause. The difference is, believers (like myself) point to design as evidence of a Designer.
And ironically, the more advanced we get scientifically, the more advanced we see the design. We see that our bodies, on a large scale, seem fairly simple... yet go down to the molecular, cellular level... and we see incredible complexity and machine-like, precision design that would have made Darwin raise an eyebrow or two
(2) The Hebrew God (YHWH) claims to be that Creator.
Many gods do not even claim that. The Hebrew God claims it, which at least means his claims are worthy of a bit more attention. I know that ardent atheists spend a bit more time over YHWH than, say, Zeus
(3) The Hebrew God has left an indelible imprint of His activities with humankind
It's interesting that we here are not really talking about Zeus, or Hermes... and yet all through history, this YHWH keeps cropping up and being talked about, even though his religion often seems on the brink of destruction at times. If Zeus were really God, I would have thought his worshippers would be making a defense of him by now.
(4) The Hebrew God has told humankind in advance what He intended to do.
For example, the "suffering servant" in Isaiah 53, who dies and comes back to life to bear the sins of many... 800 years in advance of Christ. While this could be seen as a prophecy, it is also a statement of INTENT.
(5) The Hebrew God was the God of Jesus Christ.
Jesus Christ lived and died exactly like the "suffering servant" in Isaiah 53. He also claimed to be the Son of God, and was put to death for it.
Of course, skeptics have attempted to attack and/or demolish the "Jesus" story from every possible angle...
(a) Jesus never existed, but his story is really a rehash of pagan myths and legends. ["Zeitgeist"]
(b) Jesus was mistaken. [Dawkins]
(c) Jesus was deluded.
(d) Jesus was a magician. [the Jewish Talmud]
(e) Jesus existed but his story was later edited to include miracles, etc [Skeptics]
None of these, upon closer scrutiny, hold up.
Most of the pagan myths do not bear the same resemblance to the life of Jesus, and the ones who claim this lack SOURCE MATERIAL to demonstrate this, but usually quote other writers who themselves don't have the source material to back up their claims... or where such material exists, it usually comes much later than Jesus anyway (as in the case of the supposed Mithra connections). And in the few connections, i.e. December 25th, the birthday of the Sun... well, Jesus wasn't even born in the middle of winter anyway! So the connection has nothing to do with the Jesus story in the New Testament.
Was Jesus mistaken? Possibly, but his statements do not sound like somebody who is simply mistaken. Besides, he was NOT mistaken in his pronouncements about his destiny, and the destiny of his nation. He said he was going to be put to death... and that his nation would fall by the sword and led captive into all the nations. Clearly he was not mistaken there, so something more was probably going on.
If Jesus was deluded, then let's face it, he was an incredibly wise deluded man

This also means his apostles and disciples were deluded, yet the apostles also claimed to perform miracles and also lost their lives in the fervent belief they had witnessed Jesus' resurrection. Paul was also independently deluded, because he had a vision of Christ outside of the circle of the apostles... a vision that resulted in him converting huge numbers of non-Jews to Christ!
The Jewish Talmud claims Jesus [using a cryptic name for him] was a magician, justifying why they put him to death. But this demonstrates two things: that (a) they knew he existed, and (b) that the man performed things that appeared to them as magic tricks.
But if Jesus were a magician, then that would make him a false prophet... a deceiver. Yet their own scriptures foretold a "suffering servant" who would die on behalf of the sins of the people. Daniel [chapter 9] foretold a Messiah, or "anointed one", who would be cut off... prior to their city and temple being destroyed.
Given that their city and temple WERE destroyed (as foretold by their own prophet Daniel) [in 70AD]... the Jewish Messiah must, of necessity, have arrived prior to 70AD. And while there were several who claimed to be prophets and messiahs in that time, only Jesus stands out as foretelling the things that came upon the Jews, and whose words are immortalized even today.
Thus, this suggests Jesus was a TRUE prophet... and therefore not merely a "magician". Indeed, their ancient prophets also performed similar miracles to Jesus... so that same generation would have stoned their own prophets, had they lived among them!
Thus, their claim that Jesus was a magician does not hold water, because he demonstrated all the qualities of being a PROPHET... including, ironically, being put to death by his own people, as some Old Testament prophets were.
Was Jesus' life story edited to include miracles, at a later date? This means that Jesus did no miracles... but then, why would the original disciples believe him to be the Messiah and a prophet? The miracles were one of the SIGNS and proofs of his identity. Without them, the apostles and disciples would have had no compelling reason to believe Jesus' claims... and certainly not to die for him.
Indeed, why would there be ANY compelling reason for a Jew in the mid 30's of the 1st century to become a Christian, if there were no miracles? Jesus had been put to death... end of!
Yet the apostles and early disciples staked their lives and reputations on the belief that they had seen Jesus perform miracles, and had also witnessed his resurrection. This was also true of many of his earliest disciples. The Christian congregation itself was also founded on a claimed miracle... that of 120 disciples speaking in tongues to the Jews from all nations in Jerusalem. [Acts 2] If none of these things were true, wouldn't it be much more likely that the Nazarene "sect" would have simply fizzled out after a time?
And even if not, we have to ask where their momentum came from... especially when they did not have a "New Testament" to refer to!
Word of mouth therefore clearly played a part... and the authority of the apostles. But again...the authority of the apostles was an unusual anomaly in history, because most authority is wielded for a somewhat selfish purpose, while the apostles expended themselves for, and even died for, their belief in the resurrection of Jesus.
If the apostles "made up" some or all of the miracle stories... then they are liars. But then, when you read their warnings about God, they do not sound like liars but very sincere people.
So what is left? That they were mistaken? They could have been... but then, they were also able to perform miracles themselves... so if they were making that up, then they were liars. And yet they do not sound like liars. What was their motivation? Money, power, fame? Possibly... but if so, it would be the most contradictory set of beliefs ever. Their congitive dissonance would have been through the roof
Or maybe... just maybe... they were telling the truth. Maybe Jesus was who he claimed to be... the Son of God. And maybe... just maybe... the apostles were witnesses of him, and his resurrection.
This is the best hypothesis in my opinion, and the one that I accept.
(Of course, what I've said here isn't a COMPLETE account of why I believe what I believe, or a complete attempt to refute all possible counter-arguments... but it's a good outline and a decent start

)