There is a lot of factors that go into this. Most of the time, two individuals will call on the family member. One will usually be a chaplain of the religion of the service member. The other will be a Casualty Assistance Officer. (different services have different names.) The CAO has taken a class and been trained in the job. Usually the CAO will be a senior enlisted (E-7 and above) or an Officer. Usually, the rank of the CAO delivering the news to the family will have a rough correlation to the rank of the deceased.
The remains will be shipped to wherever the family wants them to go. When coming from overseas, they always past through Dover AFB. That's where all peacetime and wartime remains are processed. <
http://en.wikipedia.org/wiki/Dover_AFB > They do all the paperwork and make sure that the remains are properly dressed and have the proper honors. The family can meet the casket at the base but that is usually not recommended. Once all the remains are processed at Dover, then they are flown to the funeral home requested by the family. The remains are also usually accompanied by a military member from the deceased's unit.
The time it takes for the remains to return CONUS depends on the rules of the country they originate from. Usually a week is a pretty good time frame, but it can take longer depending on the situation. They try to expedite the process for obvious reasons.