How did Harry know that his mother's Patronus was a doe?

He Doesn't

Harry has no specific proof that Lily's patronus was a doe. He's basing that assumption purely on the fact that HIS patronus is a stag, which is based on his father. (Lupin points out the similarity in Harry Potter and the Prisoner of Azkaban) He assumes Snape's patronus is Lily/Lily's patronus because, to Harry, his Father's deer Patronus would be matched by his mothers, who being a woman would have a female deer (ie: doe). Once he relives the famous "Always" scene, he assumes that, like Tonks, Snape's patronus takes its form based on Lily's, whom Snape still loves. Apparently Rowling has confirmed Lily DID have a doe patronus on Pottermore, but there isn't any proof given in the books or the movies.