Answer:
It depends on where you are. In some locations, no specific education or training is required; some elected official (such as the district attorney or sheriff) is automatically also the coroner. In other locations, the coroner position is a licensed physician, and in some in addition to being a physician the coroner must specifically be a forensic pathologist.
In places where the coroner is not required to have medical training, the position is more of a bureaucratic position: the coroner is essentially just a keeper of records that relies upon the expertise of medical professionals. In those where the coroner is a physician, the coroner probably actually does at least some of the determinations of things like the cause of death personally.