So, more specifically, where did the information or traditions about early Christian martyrs come from?
Obviously the Bible has St.Luke's Gospel and then the Acts of the Apostles, researched by the early church historian St. Luke, but what about the traditions on the rest of the disciples, or most of Foxes book of Martyrs?
More specifically, Peter is said to have been crucified upside-down, but why was he crucified? Who recorded the death?
Are there non-Christian sources for their deaths, or is it all tradition/later research by the church?
It depends on which specific tradition or martyr you are talking about. For Peter’s death, there is an account in the 2nd century work Acts of Peter. You can read the text here, skip down to XXXIII to read about the martyrdom in particular. Mention is also made of Peter’s martyrdom in 1 Clement, probably from the very late 1st century or early 2nd century, which you can read here, skip down to 1Clem 5:5 and 6 for the mention of his death.
On the whole, martyr traditions are generally written by Christian believers, since non-believers rarely had any impetus to record the deaths of Christians as being particularly significant.