Run this command to install MySQL server, as well as the packages for the client and for the database common file:
sudo apt-get install mysql-server
NOTE: A dialog pops up for an optional root user password. (I advice you provide one).
Run the following commands:
sudo apt-get remove --purge mysql-server mysql-client mysql-common
sudo apt-get autoremove
sudo apt-get autoclean
sudo rm -rf /var/lib/mysql