I answered a similar question recently: What was life like for common people during the crusades, especially during the siege of Jerusalem? But that's about common people in general. I can expand a bit to answer your question about the Christian population.

When the crusade arrived in 1099, the native eastern Christians in Jerusalem itself weree Greek, Armenian, and Syrian Orthodox. There may have also been Coptic, Nestorian, Georgian, Ethiopian, and Maronite Christians, although probably not in Jerusalem itself. There were also Jews and Samaritans, as well as Sunni and Shi’i Muslims.

Jerusalem had been under the control of the Sunni Seljuk Turks (ultimately ruling from Baghdad, where the Sunni Abbasid caliph lived) since 1070. Before that, it was ruled by the Shi’i Fatimids in Egypt, who had their own caliph in Cairo. In 1098, when the Seljuks were distracted by the crusade in the north, the Fatimids regained control.

According to the Persian traveller Naser-e Khosraw, there were 20,000 Christians, Muslims, and Jews in the city in 1050. Christians and Muslims mingled together in Jerusalem until 1063, when the Fatimids segregated them into separate quarters. The Christians lived in the quarter around the Church of the Holy Sepulchre, and, outside the city, they also controlled the town of Bethlehem. This is the period when the Holy Sepulchre was rebuilt, as the Fatimid caliph had destroyed it in 1009 (the crusaders rebuilt it again in the 1150s).

The Fatimids probably didn’t care very much about the different kinds of Christians as long as they were all living in one quarter, and as long as paid the jizya, the special tax for non-Muslims. The Christians themselves segregated themselves even further. The Greeks, Syrians, and Armenians shared custody of the Holy Sepulchre but otherwise they lived in their own neighbourhoods in the Christian part of the city.

In 1099 when the crusaders were marching towards Jerusalem, the Fatimid governor of the city, Iftikhar ad-Daula, expelled all the Christian inhabitants. That sounds cruel but it was probably a good idea from the Muslim point of view - the crusaders had already captured Edessa and Antioch and several other cities in the north by conspiring with their Greek and Armenian inhabitants. They would probably have done the same in Jerusalem.

Where did these Christians go? They probably went to nearby Christian villages and towns. In the countryside, Christians and Muslims tended to live separately from each other in their own villages, just like they lived in separate neighbourhoods in the cities. Greek villages could welcome Greeks from Jerusalem, and likewise with Syrian and Armenians.

But apparently not all of them were expelled. Some might have wanted to stay and defend Jerusalem, assuming that they would benefit more from Fatimid rule than Latin crusader rule. It’s also possible that some of them were killed when the crusaders sacked the city, since the crusaders they didn’t bother to distinguish between eastern Christians and Muslims, who probably all looked and sounded the same to them.

However, at least in Armenian accounts of the crusade, the Armenians (including their patriarch of Jerusalem) met up with the crusaders and joined in the slaughter of the Muslim inhabitants. The Latin sources don’t mention this though, so it might a bit of fiction by Armenian chroniclers, trying to claim some of the glory of the victory.

Sources:

Joshua Prawer, “Social classes in the Crusader States: The ‘Minorities’” and Josiah C. Russell, “Population of the Crusader States,” both in Kenneth M. Setton, ed., A History of the Crusades, vol. V: The Impact of the Crusades on the Near East, ed. Norman P. Zacour and Harry W. Hazard (University of Wisconsin Press, 1985)

Ronnie Ellenblum, Frankish Rural Settlement in the Latin Kingdom of Jerusalem (Cambridge University Press, 1998)

P.M. Holt, The Age of the Crusades: The Near East from the Eleventh Century to 1517 (Longman, 1986)

Thomas Asbridge, The First Crusade: A New History (Oxford University Press, 2004)

Adrian J. Boas, Jerusalem in the Time of the Crusades: Society, Landscape and Art in the Holy City under Frankish Rule (Routledge, 2001)