Looks like just some package name confusion. I believe in both cases, you're attempting "sqlite3" instead of just "sqlite".
A: Have you installed sqlite3 on the server?
# Note the package on Ubuntu is "sqlite", not "sqlite3"
$ sudo apt-get install sqlite
B: The PHP package is php5-sqlite
$ sudo apt-get install php5-sqlite
Lastly
To search for available packages, you can use apt-cache
:
$ apt-cache -n search php5-sqlite
> php5-sqlite - SQLite module for php5
$ apt-cache -n search sqlite
> ... other stuff ...
> sqlite - command line interface for SQLite
> sqlite-doc - SQLite documentation
> sqlite3-pcre - Perl-compatible regular expression support for SQLite
> sqlitebrowser - GUI editor for SQLite databases
> ... other stuff
Hope that helps!