Former cops and court reporter here, so not quite first hand knowledge.
Typically police officers give information out while keeping the details vague. If they think someone was a serial killer victim they'll probably tell the family that, but they're not going to share specifics. They're not going to give away anything that could compromise their investigation. Also, it changes if the someone in the family or close to the family is suspected.
As far as who would give the information? In a small county/small town that sees four or six murders a year the sheriff or the chief will probably talk with the family personally, if not them it would be their second in command. In a major police department the information will probably come from the lead investigator. In larger cities I have also seen victim advocates get involved, and sometimes they are told certain information that they will relate to the family.
It's been my experience that the closer to the murder timewise the less the family wants to know. This is generally speaking, but if someone is murdered in a robbery and the suspect is in custody the family doesn't seem to know an awful lot, and seems to learn most of their information at a trial. In my experience the more time that passes between the murder and the arrest the more the family wants to know. Typically the longer the case goes unsolved the more the family will know/discover/infer/be told.