[alexander] did he understand his legacy to be

Tiberius Gracchus has a really interesting take on Alexander the Great and the bits which touched me were:

Alexander, Green conjectures, built his empire in conscious emulation of the Homeric hero, of Achilles and Hercules with whom he identified and whose blood he beleived that he bore within his veins.

... this:

He showed little concern about who would succeed him- he didn't marry until he reached India and even then waited before he had children. Alexander was always planning further conquests

... and this:

But such an empire was achieved at the price of permanent instability - no sooner did Alexander leave a region than revolution sprung up behind him. Even his own appointed satraps might swiftly rebel as did Harpalus, his own treasurer, in Babylon towards the end of his reign.

So where does the Hellenist legacy fit into this scenario? Tiberius answers this and more. Have a look.