Copy all of the files to the new folder.
For the database, you’ll need to use something like PHPMyAdmin to export the current database as a .SQL file.
Edit the .SQL file you downloaded to remove the first statements specifying what database to use (it will be “USE database”).
You’d then need to create a new database using your website’s control panel software (like cPanel of Plesk for instance).
Then go to PHPMyAdmin once more, select your new database, and then do an Import, and select the edited SQL file as the file to import.
Then edit wp-config.php in your subdirectory, to use the new database name.