Snape hated Sirius mainly for bullying him.
The most likely reason that Snape hated Sirius is because Sirius bullied him relentlessly while they were attending Hogwarts. Though it’s possible that Snape, like most everyone else at the time (including Order members) thought Sirius was responsible for Lily’s death and hated him for it, Snape’s hatred for ...
It's possible, but not really provable.
We can safely assume that Snape already hated Sirius prior to Voldemort killing Lily.
For instance, in Chapter Twenty-One of Prisoner of Azkaban we find the following:
"He hates Sirius," Hermione said desperately. "All because of some stupid trick Sirius played on him –"
This trick was to have Snape end up in ...