If Jesus Christ as was written of in the bible appeared on earth today,
Let's start here, in the middle of this sentence.
Taking a look at the character as he was written in the bible will require a lot of religious assumptions to be removed, but sure I'll be happy to roll through this thought exercise given these parameters.
As he was written to have appeared was basically a back woods redneck from out in the sticks. He wasn't at all cultured, and that was a huge sticking point for the pious religious people of the time. He came from a dirt poor village from dirt poor people. He would be much more likely to look like a refugee from a third world country to our eyes than the blonde haired blue eyed movie star hippie we associate him with in paintings. He probably looked extremely Mediterranean, if not Arabic.
He would probably look a lot like this dude.
how could he prove himself to be the son of God in your eyes?
Another interesting point that a lot of people miss in the texts is that he had an aura about him that people were drawn to. They saw / felt / sensed a holiness about him. People were constantly calling him "Rabbi" upon meeting him, and inviting him to dinner at their homes. People met him and said, "You're really cool. I need my family to meet you so I don't have to explain this thing about you that I don't have words for." People gathered by the hundreds, sometimes thousands to meet him because of the word of mouth accounts of his holiness. It exuded from him, if the way he was written is to hold any validity.
Yet another interesting point was that he didn't go around claiming to be the son of God. He wasn't boastful of himself. He talked about "God the Father", but the Jewish people refer to God as Abba which means father. This wasn't out of the ordinary, nor an implied claim to be the messiah. Most of the time when it came up, it was because people put it together and asked him if he was. So, I doubt that he would try to prove who he was to anyone if he appeared today. He'd let his message speak for itself and take root in people as truth, or move on. He wouldn't have the ego of religious people to force others to believe, just love for anyone who was open to what he had to say... because as he was written, he didn't have it then either.
To prove it to me, he'd have to have that presence, which entails a whole lot of specific traits that can't really be faked into a whole package of a person, and he would have to let me put it together myself, because as he was written, that's his style.
If he did a couple of magic tricks as he did supposedly did in the past, would that be enough?
All of us are now entitled to perform those same miracles through the holy spirit. At the last supper when he announced he'd be leaving, the apostles asked what they would do without Jesus and his powers. He said "you will do far greater miracles than I have", and after Pentecost they did, according to later accounts in the bible. Peter's shadow passing over people could heal them, according to the writer of Luke. In Acts, it is said that the Apostles healed by the thousands.
In my own life, I've seen people perform miracles in "that defied reality" kind of ways, but I've also seen miracles performed in "maybe that wasn't a miracle" kinds of ways that still defied anything but a whole lot of long shot overlapping coincidence. Miracles would impress me, but they wouldn't be proof. They would just be another factor for him to be 'as he was written'.
The point to Jesus being on Earth was to be an example of what we can do when we have a relationship with God. He never did anything a normal human being can't, aside from being the lamb of redemption. If he wasn't a normal person while he was here, it would just be God showing off. His miracles weren't to show off. His miracles were to show us that we can do them, and how amazing a relationship with God can become. His perfection came at the end, because nothing can be perfect until it is complete. "It is finished."
Therefore, whether or not he performed miracles would not be a factor in whether or not I believed he was who he probably wouldn't even say he was.
Once he proved himself, if he told you the bible was false, how would you feel about that?
That's essentially what he did in the New Testament about the Old Testament. That's why they're different books. Part of his purpose was to explain how they'd gotten it wrong, clarify, and fulfill. He upset a whole lot of religious people by telling them that what they believed was wrong. However, he never said the bible was false, so I don't see him taking that approach in a modern incarnation. He'd very likely take up the cause of reforming and telling people where they got it wrong, and teaching his original way, which we've diverged from in the past 2,000 years.
If I met someone who had that aura of holiness, performed miracles, and had a divine wisdom that exceeded the natural, I'd follow them out of sheer curiosity to learn from them. If at some point I pieced together who he really was, without him telling me, I'd have little reason not to believe it, because at that point he would have lived up to the criteria of "how he was presented in the bible".
But, he would have earned my trust as a teacher long before that for the reasons mentioned above, so it wouldn't change much other than my level of respect.