Hi Porty,
My initial thoughts that the local newspaper would be the Kilsyth Chronicle, but after having done some 'Googling', I discovered (at North Lanarkshire Archives in Cumbernauld):
"Kilsyth Chronicle newspaper is available to view on microfilm from its beginning in 1898 to 1960 and from 1986 to 2008. "
What came before that, I don't know. Perhaps an e-mail to North Lanarkshire Archives will give you the answer?
mcgheewi[at]northlan.gov.uk (replace [at] with @)
Anne
P.S. Did you mean 1960? Death Certificates didn't exist pre-1855