The answer lies in the Scriptures of Delphi, specifically in the collection of writings known as "The Vampire Bible".
The first vampire started out as not a vampire at all, but as a human man named Ambrogio. He was an Italian-born adventurer who fate brought to Delphi, in Greece. You can read the full story here, but in a nutshell a series of blessings and curses transformed this young man into history's first vampire.
Specifically, it began with the sun god Apollo (Greek mythology), who in a fit of rage cursed Ambrogio so that his skin would burn should it ever touch sunlight again. Ambrogio's bad luck followed when he ended up gambling away his soul to Hades (Greek mythology), the god of the underworld. The next curse came via Apollo's sister Artemis (Greek mythology), the goddess of the moon and hunting, who made it so that Ambrogio's skin would burn if he touched silver.
The blessings came soon after when Artemis, taking pity on the poor young man, gave him the gift of immortality. He would carry his curses - his skin burning by sunlight or silver, but he would live forever in his current form. Not only that, but Artemis also gave him the speed and strength to become a hunter whose skills were second only to her own.
Blood-sucking (which, by the way, is called "hematophagy" in case you were curious) is also included in this "blessing". In the vampire origin story, Ambrogio hunts swans and uses their blood as ink to write love poems to his lady Selene. While this may be considered a little creepy by our standards, it wasn't all that unusual in ancient Greece to make do with what you hunted.