Little is known, and much is conjecture ( and most of that wrong).
The Khazars were a Turkic people who fashioned a multi-ethnic Empire between the Black and Caspian Seas and for a period occupied territory to the north of the two seas as well. The Khazars were originally, as with all Turkic peoples, nomads. Their empire lasted from the 600s to the end of the 10th century, when it was finally conquered by Kievan Rus. During the time of their empire they fought against many powerful neighbours: Arabs, Byzantines, Russians, and fellow Turks. Some pockets of Khazar society remained after their defeat at the hands of Kiev, but today they have merged with all the other peoples of the region.
Our major problem with gaining an understanding of these people is that they left no written record and the archaeological and numismatic evidence doesn't take us very far.
A story has circulated for centuries, now largely disproved, that the Khazars converted to Judaism (or at least their elite did). This gained wide circulation after the publication in 1976 of Arthur Koestler's 'The Thirteenth Tribe'. Professor Paul Stampfer of the Hebrew University of Jerusalem has effectively shown such a story to be untrue, as he himself has said, that while it is a splendid story, it never happened. In an article in Science Daily, Professor Stampfer is quoted as saying, 'We must admit that sober studies by historians do not always make for great reading, and that the story of a Khazar king who became a pious and believing Jew was a splendid story'. He is further quoted as saying,'There are many reasons why it is useful and necessary to distinguish between fact and fiction - and this is one more such case'.
The written evidence in support of a Jewish Khazar state is all very dubious indeed, and totally unreliable; there being no Khazar source. More interesting are the archaeological finds and depictions of candelabra, but as Professor Stampfer reminds us candelabra were not a specific Jewish thing at this period. They merely symbolised light. Research on Khazar coins has turned up one intriguing coin which contains the words 'Moses, messenger of God'. One would have expected it to read Mohammed rather than Moses. But, as Stampfer points out, if this was a Jewish society one would expect to find nearly all the coins to bear a Jewish message or symbol. They simply don't. Thus the explanation for this one coin is that the mint master was a Jew, a profession practised by Jews across Europe, and Stampfer suggests this may simply amount to an anti-Islamic message by countering it with a pro-Jewish [even perhaps a pro-Christian] one.
Maybe one day archaeology will provide final proof, but to prove a negative, ie the Khazars were not Jewish is far more difficult than proving a positive, ie in this case the finding of a Khazar synagogue. Sadly for those of us who like good stories, this story has all but run out of legs.
The final Jewish connection with the Khazars has been the belief that they became the direct descendants of Ashkenazi Jews in Eastern Europe. This we can confidently say is 100% untrue as DNA research in recent years shows no possible match.
This whole early medieval period is murky and nowhere murkier than with the story of the Khazars.
As for further reading we shall have to wait until the contemporary research into the link between Khazars and Judaism is written up in popular form, for all the earlier, and indeed quite recent books, perpetuate the myth.