Who and what were the 5 Patriarchs? Early Christianity had a huge variety of splinter sects, many of them teaching ideologies that modern Christians would hardly think of as "Christian" at all, but most of the "Gnostic" sects were quite small, a teacher and a handful of disciples, and this was by design. The Gnostic marketed themselves as for the elite, imparting secret wisdom beyond the grasp of most people. The katholikos "for everybody" Christian sects (the term orthodox "right-believing" did not start to be used until the 5th century) aimed at the masses, and indeed were often criticized for appealing to the vulgar and the gullible. Communications were sporadic, and significant regional differences in beliefs as well as practices did emerge. But five units more-or-less recognized each other's legitimacy, and were formally declared administrative divisions of the Church by Justinian (but the Pope at this time already refused to accept a Church council ratifying this, claiming that he was not just one of five co-equals). Roughly in order of age, these were: the Ebionite church of Jerusalem; the Gentile church of Antioch; the Pauline churches headed by Rome; the Johannine churches eventually headed by Constantinople; and the Coptic church of Alexandria.
The church in Jerusalem, despite its obvious seniority, did not maintain a leading role because it early went off in a very different theological direction from the others. Once Peter and John had left, it was under the sole direction of James the Righteous, brother of Jesus, and after his murder the succession passed strictly to the next-of-kin for a century or so. They were called the Desposyni "royal heirs" (despos "king" in Greek did not have the derogatory overtone that "despot" has since acquired), each of them in turn being not just a "bishop" but a "Messiah", that is, anointed rightful king. However, Jesus to them was not just one king among the others in the lineage, but the "king of kings" (a Persian-derived title) who would judge all kings, as well as all other men, at the end of time. He was, however, the most-exalted human, not a God: in particular, the Ebionites denied the "virgin birth" story, which was somewhat embarrassing (who would know the facts better than the family?) The "Protevangelium of James", a late forgery purporting to tell about the life of Mary and the virgin birth, is ascribed to James the Righteous for the propagandistic purpose of claiming that even James believed that story.
The Ebionites had their own version of the "Gospel of Matthew" in Hebrew, which was seen by Jerome and Epiphanius, who note some of the major differences between it and the Greek gospel (unfortunately, we don't have a good text of it). It contained the "Q" material (in the original Hebrew; this is what I think Matthew actually wrote) combined with the Markan narrative (translated into Hebrew), but none of the "F" material (the opening genealogy and nativity story, and other insertions with scriptural quotes and their "fulfillments") or "P" material (a couple stories emphasizing that Peter was #1 plus the ending with the "guards at the tomb"). Instead there is an ending where the risen Jesus meets brother James and shares a meal of bread, fish, and wine with him. This appears intended to emphasize not just the primacy of James, but also that the risen Jesus was just as corporeal as before, not a "glorified" body of some spiritual substance as in the Pauline epistles.
Eisenman has written an informative and highly irritating book "James the Brother of Jesus". He dumps a huge amount of enlightening data about the historical context, in a jumbled order because he keeps losing the thread of his argument to chase down tangents. He filters everything through an intensely Jewish-nationalist and anti-Pauline lens. He wants James and Paul to have been mortal enemies; more accurately, it seems that their relations were often strained but generally polite and correct. He wants the Ebionites to be the same thing as the Zealots, but while there were clearly connections, info that Eisenman himself brings out tends to show that James was very averse to creating any trouble with the Romans. He is even more cynical than I am about the canonical NT, always asking what propagandistic purpose is being served by any piece of text and speculating about what less-pleasant truth might lie underneath; but takes the Ebionite literature (of which he shows that we have more surviving than you might expect) at face value, without any of the same kind of scrutiny. He does make it clear, however, that James was seriously de-emphasized by the NT, and was actually a very prominent figure in mid-1st-century Jerusalem.
His sermons were called the "ascents of James" because he would climb up to the ramparts of the Temple, to address as wide a crowd as possible in a booming voice, wearing a high-peaked white crown (we now associate such a miter with "bishops" but it was the Mideastern style for a "king"). His focus was largely on Jesus, reminding the crowds of how his brother had been executed for no crime but speaking truth, how he had meekly submitted to show that he knew that what is truly of value is not in this world, and how he had risen from the dead as a sign of God's singular approval of him. He called for "righteousness" by which he meant thorough observance of the Torah; but always interpreted Torah as serving the twin principles of honoring God and serving the needs of our fellows, rejecting any legalisms which did not serve such purposes. In this he was not alien to the teachings of Hillel. The Epistle of James in the NT is written in better Greek than it is plausible for him to have known, but may have been written by a secretary at his dictation, against an extreme "salvation by faith alone" interpretation of what Paul was teaching at the time; even if it is actually from a later follower, however, it seems to be true to James' position.
One cannot help but ask how he got away with this, for decades. The priests in the Temple cannot have liked constant reminders about the Jesus case, and you would think the Romans would be alarmed by a "king" openly appearing in a crown. But unlike Jesus, who often made trouble (disrupting the animal sales at the Temple, flouting Sabbath rules), James was a punctilious observer of the Law and gave no excuse for accusations, and he seems to have had sympathy from the people, even among those not inclined to convert to his beliefs about Jesus. And he made plain that his kingship was a "spiritual" one, that is, that his role was to call people to righteousness, not to compel them. And, he acted "as if he owned the place" because, technically, he did: David's purchase of the Temple grounds was recorded in scripture; Zerubabbel had built the second edifice there; and the property had never been alienated from the family of which he claimed to be the heir (and nobody seems to have disputed him on that).
He was at his gravest danger, not from the Romans, but from Herod Agrippa, whose double descent from the Hasmonean and Herodian lines led him to think he could break off from Rome, forming a middle-state between Roman and Parthian Empires. Agrippa's negotiations with Roman and Parthian vassals were well advanced, and apparently he was about to proclaim himself as an emperor (of a realm including Egypt, Palestine, Phoenicia, Syria, and Mesopotamia) when he mysteriously died (struck down by God, or maybe just poisoned). During this time, Flavius Josephus reports that Stephanos "Stephen" was stoned to death by a mob of lestae "bandits" for being "a servant of Claudius Caesar"; lestae is a common derogatory term for Zealots, used also in Josephus for the young bucks who attacked Pontius Pilate's new aqueduct ("what have the Romans ever done for us?") to deface the Roman eagles Pilate had dared to decorate it with, and it is possible that the Barabbas released instead of Jesus, described in the gospels with the same derogatory term, was one of the ringleaders of the aqueduct raid. If this is the same Stephen, which seems a reasonable assumption, the Jewish nationalists were offended at the Christians principally because they opposed Agrippa's project of breaking away from Rome. Around this same time, according to the Ebionite literature, James was knocked down off his perch, twisting his ankle so that he could not resume preaching for a while, by a mob incited by someone who is vaguely described, but one manuscript flat-out says in a marginal note that it was Saul of Tarsus. The canonical Acts says James son of Zebedee was beheaded by Agrippa, and Peter arrested, although he doesn't seem to have dared to arrest James the Righteous.
James survived twenty more years. Jewish/Roman relations got worse and worse: one year a soldier guarding the Temple during the Passover rush mooned the crowd, who threw stones at him, starting riots in which many lost their lives; a group of radicals repeatedly obstructed the delivery of Rome's offerings to the Temple (the imperial government donated to all religious institutions) which they thought the priests should never have been accepting; then the Roman governor was assassinated, with the rumored connivance of the High Priest. On Yom Kippur, James went into the Holy of Holies to make atonement for the sins of Israel; this should have been the High Priest's role alone, but it was the High Priest's sin above all others which needed atoning for, and James apparently felt that in such a case it was up to the King. The symbolic repudiation of the High Priest's legitimacy was a step too far: taking advantage of the vacuum in the Roman administration (the replacement governor was still en-route; it had taken time for word to get to Italy and back), an irregular Sanhedrin was convened, condemned James to death (they were not supposed to do that while foreigners still occupied), and a mob knocked James down and beat him to death with clubs. Later authors sometimes said that this murder (not the killing of Jesus) was the particular sin which called down on Judea the wrath of God; the spiral of violence that ended with the destruction of the Temple escalated from this point on.
Under Domitian, two young Desposyni were captured and questioned about being the heirs to David; they played dumb, saying they were just humble peasants, and got away with their lives. Under Hadrian, the second Jewish Revolt, under Simon bar-Kochba (who "claimed to be somebody" but not an heir of David), led to a ban on Jews living anywhere within a wide radius of Jerusalem, renamed "Aelia Capitolina" with a temple of Venus (complete with sacred prostitutes) on the Temple Mount. The Ebionite headquarters was mostly at Pella in Transjordan for a while; Christianity remained a "Jewish sect" in the eyes of the authorities of "Syria Palestina" as Judea was now called, so they were doubly illegal there, regardless of the fact that the Jews now reckoned the Ebionites among the minim (precise meaning of this word not known) who were not to be allowed to enter any synagogues. But much as Christians elsewhere in the Empire were not actually persecuted full-time, the enforcement of the law against Jews in Judea was very spotty; by the time of Constantine there were large communities of Jews and Ebionites right in Jerusalem.
According to Epiphanius, by the 3rd century the Desposyni were extinct, and there was a division between "Ebionites" who rejected the virgin birth and the Pauline epistles, versus the "Nazarenes" who accepted both (the two words had earlier been used as synonyms). There was also a Gentile Christian church in Jerusalem, but the leader called himself the "bishop of Aelia" using the official name of the city (rather than the name that just about everybody still called it), and was answerable to the metropolitan of Caesarea ("metropolitan" is used for "archbishop" in the East; Caesarea was a Roman-founded port, majority-Greek even before the first Jewish Revolt), who answered to the patriarch of Antioch. It is unclear when the line of bishops of Aelia started, because they had to be even more underground than most Christians at the time. So, after Constantine one of them, named Makarios, "discovered" that actually his line stretched all the way back to James the Righteous, and they became "patriarchs of Jerusalem". They had difficulty with their congregants: a successor to Makarios, Melito of Sardis, complained that even those with no Jew in their ancestry were mostly Ebionite in sympathies. But the Muslim invasion (bloodless here; the Byzantine government had lost support thoroughly, and the Muslims were welcomed) solved this, Christians of Ebionite leanings simply converting to Islam (whose view of Jesus was not alien to their own), leaving the patriarch with only the orthodox.
During the Crusades, a new line of patriarchs of Jerusalem loyal to the Pope was appointed, and the Armenians who had a lot of monasteries around Jerusalem did not see eye-to-eye with either Rome or Constantinople and have their own patriarch. Saladin welcomed all the varieties of Christians and Jews back to Jerusalem (not just enlightened tolerance, but also sound business: the pilgrimage trade was a big money-maker). The Latin, Greek, and Armenian patriarchs were constantly at odds over the Holy Sepulchre, and begged Saladin to appoint a neutral party (Muslim, of course) to hold the keys. The descendants, the Nusreibah family, still have "the keys" (the physical keys of course are totally symbolic now, the locks to which they went having long since rusted away). The current key-holder tried to run against Hamas and Fatah in the ill-fated Palestinian elections, but got only a small share of the vote. He is reviled by Palestinians as "collaborationist" for urging a renunciation of violence, and by Israelis as "intransigent" for continuing to demand that East Jerusalem be the Palestinian capital. The best example of the kind of nonsense his family has had to put up with over the centuries is a ladder left behind when someone tried to do some overdue repairs, causing a furore because all factions had not been consulted, and since no agreement could be reached over which faction had the right to remove the ladder, it remained for centuries until finally removed by natural processes of decay.