how to search for a particular string from a .gz file?

You can use zgrep. Usage is similar to grep.

zgrep "pattern" file.gz

From the man page’s description:

Zgrep invokes grep on compressed or gzipped files. All options
specified are passed directly to grep. If no file is specified, then
the standard input is decompressed if necessary and fed to grep.
Otherwise the given files are uncompressed if necessary and fed to
grep.