This is a fascinating topic, but we don't have much to go on in the way of sources. The classic source on the conversion is in Simon Dubnow's History of the Jews in Russia and Poland: http://www.gutenberg.org/files/41547/41547-h/41547-h.htm#Page_19 . He recounts the popular tale of the Khazar king receiving delegates from Christianity, Islam, and Judaism in order to make a decision about which religion to adopt.

The only primary sources we have are the letters of Hasdai ibn Shaprut to the Khazar king in the 10th century. You can find those here: http://www.fordham.edu/halsall/source/khazars1.asp .

We know that the kingdom existed and that the aristocracy probably converted to Judaism, but not much beyond that (the vast majority of the population probably did not convert). It is really not clear why the kingdom chose Judaism apart from a political desire to remain free of Byzantine or Abbasid influence.

There is significant doubt that the Khazar conversion ever took place; Shaul Stampfer at the Hebrew University of Jerusalem wrote an article I recall reading (or reading about?) earlier this summer finding no convincing evidence that the Khazars ever converted. I admit that this is far beyond my expertise or usual area of interest, and we may have an expert on the history of Judaism or the region who can say more or perhaps offer a counter to Stampfer. Still, information regarding Stampfer's argument is available here: http://new.huji.ac.il/en/article/22007