Got a death certificate for a friend three years ago, your situation will be different but it might help with the procedures.
Good friend living in Chiang Mai and died in Chiang Rai, partner and son have moved to Europe at that time.
First thing was to get hold of his passport that he did not have with him so he could be officially ID'ed.
To get the death certificate I needed three other documents.
He died in the Overbrook hospital, a private hospital in CR and I got the form where the responsible doctor declared him dead, the body was already moved to a bigger government hospital where an autopsy would be performed if there was any doubt about the cause of death, they issued a statement with the official cause of death and that nothing suspicious was found.
Next was a visit to the police and they wrote a police report.
With these three documents I went to the District Office, Amphur, and they wrote the official death certificate.
There had been a couple of phone contacts with the Embassy and they were helpful.
Of course this should be translated and legalized but that is not difficult to do, most translation services will do that for you.
As both his and my name are on it in Thai script you will have to tell the translator what must be the outcome of the name translation otherwise you and up with quite a different name and it will not be accepted.
Joop