I answered a similar sort of question a few weeks ago here.

The tl;dr is basically that you succeeded as local priest (as the parochial system wasn't very well developed until the later Middle Ages) if your dad (or uncle, grandfather, etc.) had been the priest before. I should add the important caveat that this really only applies to the end of the Early Middle Ages (from about 800 onwards) and only really in Western Continental Europe. In earlier periods, I don't really know, but hopefully a late antiquist can come along and clear things up. If there are any follow up questions though, I'll do my best to answer them.