Good question! The short answer is, we don't know for sure, because it is never explicitly stated.
The slightly longer answer is, there are some clues in the poem itself we can use to draw tentative conclusions. Look first at the complexity of the speech. This is definitely an educated speaker. By definition, then, it is not one of the "mute inglorious Miltons" buried in the graveyard. It is also not the "plowman" or one of the fellow villagers. They don't have the psychic or psychological distance to see things from this point of view, nor, to be blunt, enough energy. They've been farming all day, and they are tired.
Given the setting (a churchyard), the language, and the perspective, I say we have two or three options.
1) An educated outsider, like, well, Gray, especially a Romantic poet given to seeing the rural settings as poetic.
2) A dead person. (It is a churchyard, after all.) I think this less likely.
3) Some supernatural being (again, churchyard, and sees things no one else does). This seems unlikely as well, because the speaker has questions, not answers.
So, I'm going with the limited answer of #1.