During the First Crusade, the Byzantine Emperor Alexius I provided extensive support to the Crusaders, but with ulterior his own motives: his intent for the Crusaders was that they would retake the portions of Asia Minor that had been taken from the Byzantines by the Seljuqs. This is why, in exchange for his support, he made the leaders of the Crusader armies agree to return any captured cities that had once belonged to the Byzantines to him upon their seizure of it.

(In point of fact, the agreement was that they would return to him any city that had ever belonged to the Roman Empire, which theoretically included Jerusalem, but by the time they took Jerusalem, relations between the Crusaders and the Byzantines had deteriorated considerably.)

The Byzantines generally did not commit actual troops to the First Crusade, but did provide material support and advice. The main exception was during the siege of Nicaea, when Alexius sent a small fleet overland to Lake Ascanius to blockade the city.

During the siege of Antioch, Alexius became convinced that the Crusaders would fail and abandoned them. This considerably damaged future relations between the two groups, as you can imagine.

Of course, the Byzantines weren't the only eastern Christians in the region. The Levant had a significant Christian minority, mostly of Armenian ethnicity. The Crusaders took extensive advantage of this opportunity, and the local Christians did provide much advice on fighting the Fatimids who controlled Jerusalem, including the secret to defeating Greek fire (which was that it could not be doused by water, but could by vinegar).

Source: The Crusades by Thomas Asbridge