There is evidence outside of Scripture that confirms the Christian faith. However, you have to understand that records, contemporary to Jesus's time, are very fragmentary and few have survived. Remember that the papyrus and parchment, they were written on, only lasted a short time. Without someone to copy them — laboriously, by hand — onto new materials over and over again throughout the centuries, many simply disappeared. We don't have much confirmation of anything that happened during that period. Remember, this was long before the invention of the printing press in the 15th century.
There are several different kinds of evidence that we do have available.
First of all, there are secular references to Christ's existence, for example, in Flavius Josephus (Antiquities of the Jews). Wikipedia (Historicity of Jesus) notes,
"The Roman historian Tacitus, in his Annals (written ca. A.D. 115), book 15, chapter 44.,  describes Nero's scapegoating of the Christians following the Fire of Rome. He says that their founder was named Christus (the Christian title for Jesus), that he was executed under Pontius Pilate, and that the movement of his followers, initially checked, then broke out again in Judea and even in Rome itself."
If you're merely trying to establish that Christian doctrine was believed by Christians from sources outside the Bible, there are several Christian sources that bear testimony to Christian belief early on. These are often online. See:
. . . and can easily be read. Probably the earliest source is the Didache, or Teaching of the Twelve Apostles (first century). This is a basic manual for a Christian believer and is very valuable as such. It gives, for example, instructions on Baptism.
Another early one, probably before A.D. 70, is the (First) Epistle of St. Clement of Rome to the Corinthians. St. Clement, the bishop of Rome (i.e. the pope), rebuked the Corinthians for overthrowing their priests (presbyters).
There is also:
That's just the first century. Even richer are the second century Epistles of St. Ignatius of Antioch (extensive) and the Epistle of St. Polycarp of Smyrna to the Philippians.
Also in the second century are the writings of St. Justin Martyr and St. Irenaeus of Lyons. These are theologically much deeper. There is, basically, no lack of testimony from Christians themselves, even testimony contemporary to Scripture. They certainly testify to the Resurrection of Jesus. The fact that a whole lot of people were giving up their lives in the conviction that Jesus rose from the dead is pretty good evidence that He did.
There is no contemporary surviving evidence of the darkness of the Crucifixion, though some have suggested it was symbolic rather than literal. It would not have been a solar eclipse, which would have been impossible at Passover.
As for the virginity of Mary, I don't think that would be a fact likely to be attested to or verified by independent secular sources, although I do find it interesting that traditionally Mary was understood as maintaining her virginal integrity before, during, and after Christ's birth, which would have preserved objective evidence of it. The earliest Christian testimony I can find is in St. Ignatius in A.D. 107:
As for the Exodus, you have to understand that event was four thousand years ago and the chances of finding precise evidence for an event that long ago is pretty slim.
The final kind of evidence for Christian belief is simply to look at the New Testament texts as you'd look at any other historical documents — take them at face value. One interesting thing you notice is that the text does not whitewash the mistakes of the Apostles — they are often portrayed in a fairly negative light. For example, Jesus gets frustrated with them and rebukes them on several occasions. This points to the authenticity of the text — generally if you're making up a text, you make yourself look good, not bad. The gospels are brutally honest (as is the rest of the New Testament).
I would highly recommend for further reading this article: