I mean he can take control of a living brain/person. Can he totally morph into a dead brain? Can he read the memory of a dead person?
Can he control someone who was just killed? Or anyone killed recently? Or even years ago?
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Presumably not. The movie Logan provides some evidence that his control over a person stops at their death-- several people paralyzed by Xavier drop instantly after Logan kills them. (I've only seen the movie once, so I may be misremembering, but I can recall only one person staying upright after Logan had gotten to him. Perhaps the claws had missed just enough that the goon didn't actually die until moments later.)
I wouldn't say he could do that Professor X powers require a active mind to be able to access it. Its kinda like trying to access the files on a computer and it not even getting power or plugged in. being sleep or unconscious you'd still have active brain waves but dead I don't think you would at least not for long after death. In X3 he was able to go into a empty shell of a body because it was alive and a complete blank "slate."