Extensive lists of the library's holdings were compiled in the ancient world, starting with the great custodian Kallimachos. His list of library contents (called the pinakes) was 'published' about the middle of the third century BCE, probably by his pupils. Almost immediately, revisions and different versions of the pinakes were produced, for instance by Aristophanes of Byzantium and Hermippos of Smyrna. The list of Kallimachos was (apparently) the first of its kind, and its general organizational principles were copied by the custodians of other, smaller libraries (for instance at Rhodes, or at Pergamon).

Unfortunately, not a single scrap of the list survives. It is hoped that one day at least a fragment of the pinakes will be recovered from papyrus finds, but as yet nothing spectacular has surfaced (I'm confident it is only a matter of time, however -- there are over a million un-read papyri in University holdings, including hundreds of thousands (if not more) in Oxford's Bodleian holdings.

What we know of the list(s) comes from much later late-antique sources, often in generalized form.

The ultimate bible for this topic is Blum's Kallimachos und die Literaturverzeichnung bei den Griechen, which has been (thankfully) translated from the German original by Wellisch. That English translation is Kallimachos: the Alexandrian Library and the origins of bibliography (1991, U. Wisconsin Press).