How do I teach myself the Byzantine/Medieval Greek language, i.e., around the 9th century?

Hm. Noone teaches Byzantine Greek as something distinct from Ancient Greek. That’s because for most purposes, it isn’t distinct.

I’m going to go through a potted history of Byzantine Greek for others who might stumble on this question.

There are three registers of Mediaeval Greek to consider; I’ll use Mediaeval to include Greek under Latin rule.

• The vernacular doesn’t show up much at all; nothing systematic before the 14th century. There is exceptionally a vernacular corpus from the 9th century, the Category:Bulgarian Greek inscriptions. You don’t need Modern Greek to read them.
• Low literary Greek was an officialese Koine, with occasional hints of vernacular developments, and lots of Latinisms.
• High literary Greek was Atticist: it was an attempt to write in the Attic of the ancients, with varying degrees of over-enthusiasm.

If you’re going to work with Mediaeval or Byzantine Greek, you do the following:

• Learn Ancient Greek. There will be atticisms of various flavours, a bit more than in the Gospels (though some church fathers had reasonable Attic learning).
• Be across Koine.
• You won’t need to learn Modern Greek unless you are actually working on vernacular texts.
• Read a couple of histories of Greek, so you know in broad terms what’s likely to come up. Medieval and Modern Greek (9780521299787): Robert Browning, A History of the Language and its Speakers (9781118785157): Geoffrey Horrocks
• There isn’t much in terms of a grammar of Byzantine Greek; An historical Greek grammar chiefly of the Attic dialect [By] Antonius N. Jannaris is still the best out there. There are specialist articles, but you likely won’t need them.
• Bring a dictionary. Bring three, actually: A Patristic Greek Lexicon: G. W. H. Lampe; Lexikon zur byzantinischen Gräzität; Greek lexicon of the Roman and Byzantine periods (from B.C. 146 to A.D. 1100) : Sophocles, E. A.
• Be prepared for oddities. The oddities will depend on who’s writing and when. Atticists get enthusiastic to the point of fictional grammar. (When Harry Turtledove was a Byzantinist, he wrote “Most Byzantine historians felt they knew enough to use the optatives correctly; some of them were right”.) Byzantine authors systematically misaccented words, because they could. Some authors slip up and let vernacularisms in, though that’s more a late Byzantine thing.