How would a suspicious death be investigated in early 19th century California?

There were marshalls in California and local Sheriffs.

Yes, the Sheriff's Department then resembled - and was - a Police force.

Who would investigate the Sheriff's Department?  I don't understand the question.

The 19th century was a hundred years long and policing techniques changed greatly during that period. But by the mid to late 1800's San Francisco was a thriving city with its own police force. Most of the rest of the state remained less developed than the Bay Area and may have used County Sheriffs' Depts rather than municipal police forces. 

They would have just dealt with it as they do today, with whatever technology they had at the time.

What do you mean, "suspicious"?