If you’re using a version of Ubuntu prior to 19.04, you’ll need to install libmysql-java, the MySQL JDBC driver, because the connector is not in the Ubuntu-packaged MySQL package that you’ve already installed. Versions after that do not publish that package.
Once it’s installed, you’ll have the file /usr/share/java/mysql.jar, which is an indirect symlink to the actual jar file.
Alternatively, install the mysql-apt-config package, update, and you’ll be able to install all currently supported MySQL software with apt.