Whom should I ask for upgrading the particular package, if there is a new upstream release?
Should I file a bug report?
What if the package is maintained by Debian maintainers? Should I leave a comment at their mailing list?
The maintainer of the package. He might already be working on it!
How? Doesn't really matter. I see bug reports for it, but usually this is discussed behind the scenes. I guess most is being discussed using IRC and mailing lists. Depends on the package and the people working on it.
Debian/Ubuntu developers (package maintainers) are uploading new releases to
Exception: web browsers.
See also this answer on the policy regarding updates within Ubuntu.