Most of the story is told in the first person by Yu Tsun, in the form of a deposition (oral testimony given by a witness to be used in a trial) that he narrates from prison. One aspect of Borges' genius, however, is the way he forces us to consider the same story from multiple perspectives at once.
What about that confusing introductory paragraph? By including a frame to Yu Tsun's tale, we come to see his narrative as a sort of long quotation within a larger work. In fact, it's presented as a document used as evidence within a work of history. We could make the argument, then, that the super-narrator for this story is a third-person objective narrator – an academic with his own personal spin on the matter at hand.
Then there's the matter of the footnote. Who is this mysterious "manuscript editor"? Well, he's evidently British, since he's so particular about Captain Madden being a hero instead of a villain, and likely the person recording Dr. Tsun's deposition. But who made the italicized annotation "note from the manuscript editor"? And more important, why does Borges include all these anonymous interjections in the first place?
Well, one possible explanation is that it heightens our awareness of how the text is constructed. As we read, these little interruptions remind us that the story is a document that has been put together and influenced by lots of different people. When we have to contend with a whole bunch of strange authors and can't be sure that they are reliable, we're forced to be a little bit more careful about trusting our narrators.