It was almost certainly a scroll or a small set of scrolls.

Codices - books as we know them today - were an invention of the Roman imperial era. In Alexander's time, texts were typically written on papyrus scrolls. Although the dimensions varied considerably, the "average" scroll seems to have been about 12 feet long and 10 inches high. Especially when combined with large characters or wide margins, this format imposed fairly strict limits on the amount of text that could be written on a single scroll. Long works were thus divided into "books" - famous examples include the 142 books of Livy's history, the 15 books of Ovid's Metamorphoses, and (of course) the 24 books of Homer's Iliad.

The Iliad of Alexander's day was not yet divided into the now canonical 24 books; that happened after his death, in Hellenistic Alexandria. But it was, we assume, essentially the poem we know today in both substance and length. There were bulky editions of the Iliad, like the version written on a the skin of a gigantic Indian python (Zonaras 14.2). There were also novelty miniature versions, like the Iliad written literally in a nutshell (Pliny, Natural History 7.85). Most editions of the Iliad, however, probably consisted of nothing more or less than a dozen or so papyrus scrolls.

Alexander had been fascinated by the Homeric epics since boyhood. One of his first acts after crossing into Asia Minor had been to visit the supposed site of Troy and reenact the funeral games of Patroclus, and after taking Gaza he reportedly had the Persian commander dragged behind a chariot like Hector (if we can believe Curtius 4.6.29). Alexander's former tutor Aristotle had composed or commissioned a special edition of the Iliad, which the king brought with him to Asia (Plutarch, Alexander 8.2; 26.2). We don't know, however, whether this was a full edition of the poem or a "greatest hits" compilation focused on Achilles, Alexander's favorite character.

After his great victory at Issus, Alexander was presented with a jeweled casket from the treasures of Darius III. This, he decided, was a receptacle worthy of the Iliad, and the casket served thereafter as a carrying-case for his scrolls. (We don't know how large or elaborate it was, though I would guess it was a bit too bulky to fit under the king's pillow, as the scrolls themselves supposedly had.)

Whatever its precise format, Alexander's Iliad seems to have been eminently portable - perhaps only a few scrolls, traveling in the luxury of a Persian jewel box.