The Iliad was passed down through the Greek generations as an oral song/poem to be performed, but Alexander was known to never be without a written copy of it. At what point did the Greeks begin to write down the Iliad and Homer's other stories?

I'm studying a Classical Literature Module as part of my degree so I'll do my best to answer this though I'm sure someone else could provide a more comprehensive answer.

The Iliad we know was likely written down between the early 8th Century BC and late 5th Fifth Century BC. The justifications for this timeframe is as follows:

Earliest Estimations:

The Greek alphabet emerged in the early 8th Century BC
The finding of the 'Cup of Nestor' at Pithekoussai. Dated to 750-700BC it has on it an inscription beginning with "Nestor's cup, good to drink from." Opinion is divided on this, many feel it alludes to the Iliad while others claim that the cup of nestor existed in mythology outside of the Iliad.

Later Estimations:

In Book 9 (379-85) of the Iliad, Achilles refers to the gold taken from Egyptian Thebes which we know was sacked c. 663 BC.
In De oratore, Cicero claims Pisistratus (Athenian tyrant, 561-527BC) was the first to bring together Homer's "disordered books" (libros confusos) into one work. (De Oratore 3.34.137)

Here's some extra reading if you're interested:

M.L West (2010) Making of the Iliad
M. Dickie (1995) “The Geography of Homer’s World” in Andersen and Dickie eds. Homer’s World: Fiction, Tradition, Reality