I believe the conversion of the Khazar was largely due to contact with Jewish traders (such as the Rhadanites) along the Black Sea. At the height of their power, the Khazars ruled the steppes from the Caspian Sea (which the Arabs called "Bahr ul-Khazar", Sea of the Khazars) to the Crimean peninsula. They were granted by the Byzantines a gold seal equal in weight to that afforded to the Caliphate in diplomatic communications; this was the heaviest gold seal used by the Byzantine chancery, and was reserved for the Caliphate and Khazaria, meaning that Byzantium regarded these as the two most powerful states in the world (in addition to herself).
The conversion to Judaism occured ca. 800 CE, and there is much debate as to how deep it went. The traditionall view was that the ruling classes were Jewish, whereas the people practised a number of religions, including Judaism, Islam and a number of Christian sects, of which the Nestorians were likely the most numerous.
Khazaria declined due to internal strife, diplomatic isolation resulting from detriorating relations with the Byzantines, and the appearance of the "Varangian Road" which drew trade away from Khazar territory. Sviatoslav of Kiev sacked the capital city of Itil in 969, and although the Khazars still existed, their territory was gradually absorbed by the Kievans, the Volga Bulgars and other neighbouring potentates.