While you might get suggestions here, /r/Genealogy is perhaps the best venue for tips if you are interested in doing this kind of research.

Are there living children? Find his obit, find his kids, ask questions! Military.com has a Hall of Valor yhat has medal citations for some awards that mght give you clues. Web sites for ship associations where he might have served.

https://www.google.com/#q=captain+john+mitchell+usn

If you want to skip methodology!