Is the Orthodox Church of Jerusalem it's on autocephalous church or is it part of the Greek Orthodox? I ask because the title of the bishop is Greek Othodox Patriarchate of Jerusalem.
There is no the
Greek Orthodox Church, there is the Greek Orthodox tradition. The term "Greek Orthodox" applies to a number of the local, autocephalous Orthodox churches: the Church of Greece, the Church of Cyprus, and the ancient Patriarchates of Constantinople, Alexandria, Antioch and Jerusalem. Generally speaking, it refers to those local churches which were within the Greek speaking Roman Empire, whose primary liturgical language is Greek, or at least was until fairly recently, which follow the "Greek" (as opposed to Slavic) usage of the Byzantine rite, and which use Byzantine chant in worship.
So the Orthodox Church of Jerusalem can be described as a
Greek Orthodox church, but it is not part of any other Greek Orthodox jurisdiction.
Yes and no. One can speak of "the Greek Orthodox Church," consisting of the Patriarchates of Constantinople (and its dependencies), Alexandria, Jerusalem, and the Churches of Cyprus and Greece. While they do squabble amongst themselves, they also tend to act as a block. Not included in this is the Patriarchate of Antioch, which follows Greek usage but has left the "Omogeneia," and the Church of Albania, whose Archbishop, though Greek, is not a Phanariot and insists on his Church being run for the Albanians. Included are the Greek Orthodox Archdiocese of America and the Greek Orthodox Metropolis of Toronto (and the Greek Orthodox Archdiocese of Australia), which under Abp. Demetrios and Metropolitan Soterios (and Abp. Stylianos), which provide a study in contrasts. Jurisdictions such as the UOCC and UOC-USA are not included, but might be seen as auxiliary (the Albanian Orthodox Diocese of America-not to be confused with the Albanian Orthodox Archdiocese of America-might seen as a little closer integrated, especially in contrast to the Albanian Archdiocese, the Mother Church of the Church of Albania).
The Jerusalem Patriarchate is at present run by and for ex-pat Greeks, heirs of the Phanariots. A century ago, with the breaking up of the Ottoman Empire and the break down of Constantinople's authority-both secular and religious-the Syndicate of Tomb Worshippers tried to institutionalize that by not only limiting their membership to Greeks (and denying membership to the local flock, who were Arab) but also making the Patriarch answerable to the Greek Church (Constantinople, Alexandria, Cyprus and Greece)-the Patriarch Damian was seen too concerned with the Arab Orthodoox Flock of the Patriarchate. Like his predecessor, who was deposed by the above because he refused to sign off on the Pot-Calling-Kettle-Black Council of Constantinople condemning the Bulgarians.
Jerusalem is independent, as it has refused, for instance, to adopt the New Style Calendar, and in the main plays the role of the most conservative branch of the Greek Church. Besides the Arab Orthodox, its flock now includes lots of Russian Orthodox and Hebrew Orthodox (Jewish converts).
If you are coming from a background of the Vatican's communion, think of its Hispanic church, governed in many different episcopal conferences in Spain and Latin America, but sharing a common background.