It is logical that once you have enough evidence for Jesus’ historical presence on earth, the next question that comes to mind is whether He really did all those miracles which are mentioned in Gospels.
For one thing, we have four independent sources of significant length (apart from scores of other non canonical texts), which is pretty good as far as ancient accounts for a single person goes. They were written by four different writers with a clear and distinct style. But the problem seems to be that they were collected and compiled under one cover as religious texts.
There is a fundamental misunderstanding in this demand for independent material. Bible itself actually is merely a collection of independent sources. There is a tendency to look at the Bible as a single work because it is now under one cover.
For the sake of argument, let’s say the Gospels were intended religious texts because they contain narration of miracles and Jesus’ teachings about God. This means then that there doesn’t exist by definition any non-religious accounts for Jesus’ life and miracles. If any account features what He taught and what He did, it pretty much counts as a religious text. If there were a “non-religious” work with as much detail as the four evangelists include, it would be considered a religious text.
Secondly, the very fact that miracles are hard to believe by non religious people, so is their narration in any historical records. These may not satisfy the physicist or the psychologist; for the matter of that, it may not satisfy the theologian either like in case of Jews of Jesus’ time. So the historical method has its limitations, just as the scientific method in general has, when it is confronted with a phenomenon which is by its very nature unique. Attempts are made to rationalize or explain away such phenomenon and this was attempted soon after resurrection, as we find in Matt. 28:13.
"We would call a miracle an event that violates the way nature always, or almost always, works ... By now I hope you can see the unavoidable problem historians have with miracles. Historians can establish only what probably happened in the past, but miracles, by their very nature, are always the least probable explanation for what happened"--
Having said that, there are some non Christian sources pointing to the existence of Jesus and then to what He did too.
A. The Testimonium Flavianum
Some reject the strong scholarly consensus that Josephus' references to Jesus are genuine, although embellished. However, many agree with that scholarly consensus represented by such diverse and respected scholars as Dominic Crossan, John P. Meier, and N.T. Wright that the first and fullest reference to Jesus by Josephus is partially authentic, with detectable signs of tampering that can be disregarded. Josephus provides valuable, independent confirmation of the existence, life, and ministry of Jesus of Nazareth. The following is the relevant part of the reconstructed version of the Testimonium Flavianum accepted by a majority of N.T. scholars:
"At this time there appeared Jesus, a wise man. For he was a doer of startling deeds, a teacher of people who receive the truth with pleasure. And he gained a following among many Jews and among many of Gentile origin."
B. The Babylonian Talmud
Another Jewish source for Jesus' miracle working can be found in the Babylonian Talmud:
It has been taught: On the eve of Passover they hanged Yeshu. And an announcer went out, in front of him, for forty days (saying): 'He is going to be stoned, because he practiced sorcery and enticed and led Israel astray. Anyone who knows anything in his favor, let him come and plead in his behalf.' But, not having found anything in his favor, they hanged him on the eve of Passover.--Babylonian Talmud, Sanhedrin 43a.
What matters is also the estimated date of their writing.
To quote from "The Miracles of Jesus: A Historical Inquiry, by Christopher Price":
The miracle stories of Jesus originated very early, contained reports not likely to have been created by early Christians, and cohere well with the rest of what we know about Jesus and his ministry. The best explanation for this evidence is that Jesus was known during his life as a miracle worker. The uniqueness of such miracle working adds significant weight to this conclusion and leads us to the further conclusion that the feats of Jesus must have been impressive. Though, as Carrier points out, Jesus lived in a time of superstition and religiosity, his miracles are uniquely attested. No other person of that time period has anything close to the attestation Jesus receives as a miracle worker. Accordingly, even if your philosophical predispositions preclude you from believing that Jesus actually performed miracles that violated the laws of nature, it should be admitted that he performed feats that convinced his contemporaries that he did such deeds.
Then there are some tangential references to the persona of Jesus. What is relevant here is these books are that they are outside bible and written between first and second century with a reference to Jesus’ life. One example is this writer called Mara Bar Serapion. We have no idea what qualifications the writer of this letter held but he wrote:
"What advantage did the Jews gain from executing their wise King? It was just after that their Kingdom was abolished. "--Mara Bar Serapion
Then there are several Gnostic writings and gospels which invariably portray Jesus as Miracle performer. Whoever wrote these texts; the writers were consistent to write about the most common theme found in the Gospels about Jesus: that He was a performer of great miracles, yet they chose to differ with Gospels, on other traits about the persona of Jesus.