Klingon politicians never seem to be very honourable. There was probably a point when one Chancellor wanted more power and managed to remove the Emperor and his family from the picture, then seized ultimate control.
But if it were as simple as that, then said chancellor would've probably just declared himself emperor. There would have to be some reason why the High Council chose to eliminate the actual position
of a single autocratic ruler.
I would think they elevated the Council to the highest position in government as a way to prevent any single individual or dynasty from holding absolute power. I can see that serving a practical function; Klingons wouldn't like to be ruled over, so if just one dynasty/house ruled everyone, then everyone else would be constantly battling it to try to take over, and the unending warfare would've devastated the society and economy. By replacing a single emperor with a ruling council representing all the major noble families, and with a system of succession to the chancellorship that gives any sufficiently strong warrior the chance to claim it, you'd allow the noble families to feel they had sufficient status and reputation that they wouldn't need to be trying to overthrow the government all the time and said government could actually get some work done.