Interest in the historicity of Jesus began over two hundred years ago and has been increasing overall till our post-modern day, when the number of books, articles, essays, monographs and PhD theses on this subject have become truly staggering. Many approaches have been taken--from quality scholarship to the fringes--and the sheer volume of material is not easy to survey. A comprehensive answer on the historical Jesus would be impossible to deliver, but I will try to limit this to a few main points about history.
So, first, you ask: "How do historians study the events of Jesus' life, and the person he was?" Secondly, "I want to know what primary source documents, etc. say about Jesus and the person he was;" and then finally "What sort of historical methods can be applied?" Let's directly address these three.
First, historians use the primary sources, and hopefully they do so in the same manner they study anything else. They begin with a question, gather all the relevant sources, assemble data, devise a method and/or use the established historical criteria and methods of criticism, apply other fields of study and secondary historical sources as applicable, and do their best to come up with a valid and viable answer to the question.
The primary sources include the gospels, which cannot be arbitrarily excluded. As New Testament scholar Gerd Theissan says "there is broad scholarly consensus that we can best find access to the historical Jesus through the Synoptic tradition." Theissen, Gerd; Merz, Annette (1996). The Historical Jesus: A Comprehensive Guide Ehrman adds "To dismiss the Gospels from the historical record is neither fair nor scholarly." They are ancient documents in their own right and must be evaluated accordingly.
Yes there is subjectivity, but there always is. Tacitus and other Roman historians included gossip, hearsay, myth, biases and errors, and they left it for the reader to sort through and draw their own conclusions. None of it has prevented good historians from making use of their writings as historical documents. (Moses Hadas, Introduction to “The Complete Works of Tacitus, ppIX-XIX). The gospel's theological content shouldn't either.
The gospels also contain the remnants of the oral creeds from the period of oral transmission that preceded the written gospels which is valuable primary evidence as well. The creeds preserve some of the earliest reports of Jesus dating from the thirties to the fifties in the first century. There were multiple creedal formulas; the most common are Christological, but 41 creeds record what the early church recorded as the historical facts of the life of Jesus.
For example, the creed recorded in 1 Cor. 11:23ff presents a tradition that was already fixed when it was passed on to Paul by the time he wrote Corinthians. It tells of Jesus attending a dinner on the night of his betrayal. It is widely recognized as presenting historical events that trace back to Jesus.
Archaeological sources also provide some primary source material as corroboration for the historicity of details, background, setting, geography, that kind of thing, and it can help with dating.
It's true that the more contemporary a source the better, but that is not an absolute requirement--sources are not automatically rejected simply because they are not contemporary. Outside of the Roman emperors themselves, there are almost no contemporaneous sources for anyone who lived in ancient times.
Sources for anyone’s existence in the ancient world are scarce, and were often written decades—or even centuries—after the person was long gone. Jesus’ background as a peasant from a humble family, the commonality of itinerant obscure-country preachers, the scarcity of contemporary writers and their focus on Rome, the problematic nature of Jerusalem and the Palestinian area itself garnering the attention, plus all the various other problems and issues in the empire of the time, would all serve to keep the focus off of Jesus while he was alive, so there's really no reason to expect contemporaneous sources of Jesus.
Historians also look at sources within the first three centuries of Jesus. Non-Christian sources include Tacitus, Seutonius, Thallus, Josephus, Pliny the Younger and his exchange with Trajan, Hadrian, the Talmud, Toledoth Jesu, Lucian, Mara Bar-Serapion, and the gnostic sources, 'The Gospel of Truth,' 'The Apocrypha of John,' 'The Gospel of Thomas,' and the 'Treatise on Resurrection.' Some of these are hostile sources, but history takes that into consideration as well. Some are less dependable than others, but information can be sifted even from those.
The writings of the early church fathers also make useful references.
There are also lost sources that have some bearing on the historical Jesus. Justin Martyr (AD 150) and Tertullian (AD 200) both say the "Acts of Pontius Pilate" was an official document of Rome, but we have no remnant of it. (This work should not be confused with later fabrications of the same name.) Justin Martyr in his First Apology said the details of Jesus’ crucifixion could be validated from Pilate’s report to Tiberius. He also says, the location and fact of Jesus’ birth could be verified by consulting the records of Cyrenius, the first procurator of Judea. We have no extant records from Cyrenius. Origen reports on Phlegon, a freedman of emperor Hadrian born about AD 80 writing Phlegon's "Chronicles" including records of reports on Jesus. It's lost. Papias is recorded in Eusebius but his multi-volume history is lost. Not much can be said with any certainty about these works, but a historical work would note them.
To sum up, there are 45 ancient sources, including the gospels, 19 creeds, various archaeological finds, 17 non-Christian sources, and five non-New Testament Christian sources that would be considered the best sources to use for researching the historical Jesus. From these sources there are 129 discernible pieces of information concerning the life, person, teachings, death and resurrection of Jesus.
Michael Grant (1992). Jesus : an historian's review of the Gospels; Bart Ehrman's Historical Jesus; Dr. Michael Burer; A Survey of Historical Jesus Studies: From Reimarus to Wright; N. T. Wright, Jesus and the Victory of God, vol. 2, Christian Origins and the Question of God.
Secondly you said, "I want to know what primary source documents, etc. say about Jesus and the person he was;"
Amy-Jill Levine states that there is a general scholarly consensus on the basic outline of Jesus' life; most scholars agree Jesus was baptized by John the Baptist, debated Jewish authorities on the subject of God, performed some healings, taught in parables, gathered followers, and was crucified by Roman prefect Pontius Pilate.The Historical Jesus in Context edited by Amy-Jill Levine
Each of these is established by normal historiographical methodology. For example, the death of Jesus is evidenced by references to it in non-Christian and extra-biblical Christian sources, by the oldest creed in 1 Corinthians 15, by medical testimony concerning the heart wound, by Strauss’s famous critique of swoon theory, other New Testament creeds, the gospel testimonies, the worship of his followers, their founding of the church, and crucifixion itself is corroborated by the skeleton of Yohanon. Taken separately, each one on its own might be considered insufficient, but in combination, the weight of evidence removes any reasonable doubt Jesus was crucified under Pontius Pilate and died. (Habermas, “The Historical Jesus” 1996, Thomas Nelson Publishers).
Other pieces of information that are held by many scholars include: that Jesus called twelve disciples; that Jesus caused a controversy at the Temple; that Jesus was a Galilean Jew born between 7 and 2 BC and died 30–36 AD; that he lived only in Galilee and Judea. (Most scholars reject that there is any evidence that an adult Jesus traveled or studied outside Galilee and Judea). That he was from Nazareth. That Jesus spoke Aramaic and that he probably also spoke Hebrew and Greek, and that after his death his disciples continued, and were persecuted.
Some scholars have proposed further additional historical possibilities such as:
An approximate chronology of Jesus can be estimated from non-Christian sources, and confirmed by correlating them with New Testament accounts.
The baptism of Jesus by John the Baptist can be dated approximately from Josephus' references (Antiquities 18.5.2) to a date before AD 28–35.
The date of the crucifixion of Jesus was earlier than 36 AD, based on the dates of the prefecture of Pontius Pilate who was governor of Roman Judea from 26 AD until 36 AD
It is most highly probable that Jesus was a real historical figure and that many of the events associated with him did happen as they are recorded.
It is most highly probable that the disciples were genuine believers in the events surrounding Jesus such as the crucifixion and the resurrection.
There are no other likely explanations for the emergence of the church.
The notion that Jesus was an invention of the Romans, or "that the gospels were revised to appease the Romans," is one of those ideas that has not stood up to historical scrutiny. It's one of many conspiracy theories about the historical Jesus. There are several of these, and it isn’t possible to cover all of them, but as an example of what some non-historic claims about Jesus are, let's take a quick look at Joseph Atwill’s 2005, "Caesar’s Messiah: The Roman Conspiracy to Invent Jesus."
Atwill’s book is based upon what he sees as ‘parallels’ between Josephus’ and the New Testament. The alleged parallels are anything but parallel. They are subjective additions to the text that do not by any standard rise to the level of actual evidence. Why would the Flavian Emperor Titus start a new religion to subdue Jews when he had already soundly subdued them on the battlefield?
Atwell's timing is hopelessly tangled. Tacitus' comment on Nero in Annals 15.44 places Roman reaction to Christianity a decade before Atwill says Titus invented it.
Atwill’s use of "typology" by the Flavians -- who are said to have learned the technique from Judaism — is wrong. Even the ancient pagans thought in these terms so there was no need for the Romans to borrow the idea. A score of Atwill's errors are the result of not recognizing that commonality often simply reflects a commonplace--something everybody did--nothing more. One of his large errors is finding commonality in names that isn’t actually there. The name Mary was held by up to a fourth of Jewish women in the first century. Atwill’s argument that the Romans turned "Mary" into a "nickname for female rebels" is simply erroneous.
This is not an example of what what "primary source documents, etc. say about Jesus." It's a fringe theory with no support amongst scholars.
Third you asked, "What sort of historical methods can be applied?" Allow me to quote from the part of an article for Wikipedia on this subject that I wrote:
The search for the historical Jesus used textual and source criticism
originally, which were supplemented with form criticism in 1919, and
redaction criticism in 1948.
The "criteria of authenticity" emerged gradually, becoming a distinct
branch of methodology associated with life of Jesus research. The
criteria are a variety of rules used to determine if some event or
person is more or less likely to be historical. In 1901, the
application of the criteria of authenticity began with dissimilarity.
In the early decades of the twentieth century, F.C. Burkitt and B.H.
Streeter provided the foundation for multiple attestation. The Second
Quest introduced the criterion of embarrassment.
By the 1950s, coherence was also included. By 1987, D.Polkow lists 25
separate criteria being used by scholars to test for historical
authenticity including the criterion of "historical plausibility".
But historians often invent their own methods or borrow them from other fields such as sociology and anthropolgy.
[Holmén, Tom (2008). Evans, Craig A. (ed.). The Routledge Encyclopedia of the Historical Jesus. New York: Routledge]; [Criteria for Authenticity in Historical–Jesus Research by Stanley E. Porter 2004]; [Denton, Jr., Donald L. (2004). "Appendix 1". Historiography and Hermeneutics in Jesus Studies: An Examination of the Work of John Dominic Crossan and Ben F. Meyer. New York: T&T Clark Int.]