How prevalent was the use of "Greek Fire" by the Byzantines?

Excellent question!

We know of several incendiary devices used in warfare throughout the ancient and medieval world; Thucydides mentions that tubed flamethrowers were used during the siege of Delium in 424 BC, and according to Byzantine chronicler John Malalas the fleet of Emperor Anastasius I used a sort of sulfur-based mixture while putting down Vitalian's revolt in AD 515. The weapon we today identify as "Greek Fire," however, is traditionally ascribed to a man named Kallinikos, who reportedly brought the weapon with him to Constantinople in 672 AD, while the empire was still reeling from the Muslim Conquests. However, Theophanes, the same chronicler who credits Kallinikos with the invention of Greek fire, also mentions that the reigning emperor, Constantine IV, ordered the preparation of fire ships over two years before Kallinikos's arrival, and today most scholars believe that Kallinikos's weapon represented an improvement to an already existing formula.

There are four consistent qualities to Greek fire in extant literature dating between 678-1204 (First Arab Siege of Constantinople - Siege of Constantinople during the 4th Crusade):

It continued burning on the surface of the water, and could be extinguished only with urine, sand, or vinegar.
It was portrayed as a liquid.
While used at sea, Greek fire was deployed from tubes or siphons mounted on the bows of specially-designed fire ships. 4)Many firsthand accounts report smoke and loud, booming noise when the liquid left the tube or siphon

Greek fire in this form was deployed for the first time during the 678 First Arab Siege of Constantinople. It was a resounding success. The Arab fleet was routed, and, following the defeat of their land army by Byzantine forces, the Arab forces were forced to abandon the siege.

Though today Greek fire has an "invincible" reputation, that is far from the case historically. The Imperial navy suffered numerous dramatic defeats between 678-1204, including the failure of the 698 expedition to retake Carthage from the Arabs (which, incidentally, led to the revolt of said fleet and their subsequent placement of Tiberius II on the throne), the 826 invasion of Crete and the 827 invasion of Sicily by the Arabs, the 829 annihilation of the fleet by corsairs operating out of Crete, and more. Out of four major attacks by the Rus on Constantinople itself, only one was met with Greek fire, suggesting that the weapon's secret had either been lost, or that it had been deemed too valuable to deploy in each given instance on a large scale. While the weapon itself was likely inexpensive, the ships required for its use were exactly the opposite; furthermore, prior to the appearance of the Arabs, the Byzantines had been used to relative control of the Mediterranean, and aside from corsairs and pirates had not faced significant naval threat for centuries. While the appearance of the Arabs forced the Imperial navy to react, after the early 8th century they no longer threatened the capital itself. While subsequent naval challenges would appear in the form of the Rus, Bulgars, and later the Turks, no major contender to imperial naval power appeared in the Mediterranean until the Italian city-states emerged as major naval powers in the early 12th century. A sense of relative safety may have caused the imperial navy to grow rather complacent, and thus curtail distribution of the expensive fire-ships to the provincial fleets. Furthermore, the emperor was reluctant to deploy the weapon outside defense of the capital itself, and, likely as a result of the naval revolt of 698, would only deploy the weapon to commanders who could be trusted to not turn against the emperor himself. These factors potentially explain the scarcity Greek fire's use, and with it the various naval disasters of the 9th century.

With the loss of major maritime centers such as Sicily and Crete, it is likely that imperial naval capabilities further declined, and Greek fire alone would not have been enough to turn the tide. Like any secret weapon, it had the greatest impact when it was first used; shock value is an important factor in warfare, and in the face of a this new ship-borne incendiary the enemy was at a loss as to how to respond. However, Byzantium's adversaries learned quickly as the weapon's limitation became apparent; the weapon had a limited range, and could only be deployed in calm seas and under favorable conditions. Eventually, opposing navies began to simply stay out the effective range of the fire ships, and began using felt or hides soaked in vinegar to counter its effect.

The Byzantine Navy deployed the weapon from specially constructed fire ships, which were equipped with a tube or siphon mounted on the bow from which Greek fire was deployed, as depicted here in the 12th century Madrid Skylitzes. The prevailing theory is that the weapon was preheated and pressurized in the below decks of the fire ship, before being discharged through the siphon on the main deck; this is not only consistent with the primary sources, but also has the great virtue of explaining the inability of the empire's enemies to adopt its use, as it required knowledge not only of the formula, but also of the devices required for its deployment. Eventually, other incendiary devices emerged amongst the empire's enemies (all of which would eventually be lumped together under the general label "Greek fire" by the crusaders) While none of these systems would use siphons like the Byzantine, other methods of deployment included clay grenades filled with the substance, and launched via catapult or by hand.

As stated, excepting the tubes other instances of weapons meeting the listed qualities can be found throughout antiquity and the medieval era, and so the question of just what was unique about Greek fire that made it so potent has ignited a controversy among historians that continues to this day. The Byzantines compartmentalized the various knowledge required for the manufacture and use of Greek fire, in order to ensure the secret never fell into enemy hands. The composition of the mixture was a closely guarded secret; Greek fire was manufactured only in Constantinople, and was distributed to the fleet commanders in the forms of sealed jars to be poured into the machinery. The machinery itself was concealed below decks, hiding both its existence and the technique to operate it from the enemy. This strategy proved quite effective; when the Bulgars captured 36 siphons and a number of the sealed jars in 814, they were unable to ever use them in battle. Compartmentalizing the knowledge may have been the cause of its disappearance, however; someone had to know the whole secret in order to oversee the process, likely the emperor and a handful of royal officials. While we are unsure of exactly when and why Greek fire disappeared from use, it is likely that the secret was gradually lost as a result of the chaotic succession process, and Greek fire likely disappeared from use long before 1204.

Another debate has long focused on what exactly Greek fire was made of. One possible formula consists of naptha, with resin as a thickening agent, along with quicklime to cause the mixture to burn on water. Some suggest that saltpeter was also a component, but this theory is today considered rather unlikely.

TL;DR: In short, not as much as you might think; while the potential period of use was a little over 500 years, it is likely that the secret of "Greek Fire" was lost sometime before then. However, various incendiary devices were deployed by several navies throughout the Mediterranean, with limited degrees of success.