Alexander can be a given "Hebrew" name (not just an adopted or given secular name in addition to a Hebrew name). In this case since his name in Hebrew was Eliyahu it appears Alexander was taken on as a secular name.
The history of the name being given to Jews is fascinating, here is something from Aish.com explaining it.
This name originated with Alexander the Great, King of Macedonia. The Talmud relates that upon meeting the High Priest of the Temple in Jerusalem, Alexander dismounted and bowed to him. (Alexander rarely, if ever, bowed to anyone.) Alexander explained that he had seen the High Priest in a dream, which he interpreted as a good omen. Alexander thus peacefully absorbed the Land of Israel into his growing empire. In tribute, the Sages decreed that Jewish boys born that year (333 BCE) be named Alexander. It remains a popular Jewish name today. (variation: Sender)"