These are the steps to upgrade drupal core.

~# cd /var/www/website

    sed -i "s/\$update_free_access = .*/\$update_free_access = TRUE;/g" ./sites/default/settings.php

~# drush en update

    update is already enabled.                          [ok]
    There were no extensions that could be enabled.     [ok]

~# drush up

    Update information last refreshed: Fri, 26/02/2016 - 08:21
    No code updates available.                          [ok]

~# drush up drupal

    Update information last refreshed: Fri, 26/02/2016 - 08:21
    No code updates available.                          [ok]

Let’s refresh the available releases

~# drush rf

Check for new core releases of Drupal

~# drush pm-releases

     Project  Release  Date         Status   
     drupal   6.38     2016-Feb-24  Security

Try to upgrade

~# drush pm-update drupal

    Update information last refreshed: Fri, 26/02/2016 - 08:21
    No code updates available.                          [ok]

Let’s Force upgrade Warning !!! backup your database, webroot and files .htaccess and robots.txt.

~# drush pm-update drupal-6.38

     Name    Installed Version  Proposed version  Message                     
     Drupal  6.37               6.38              Specified version available 


    Do you really want to continue? (y/n): y
    Project drupal was updated successfully. Installed version is now 6.38.
    
    Do you wish to run all pending updates? (y/n): y

    Executing system_update_6056                                            [success]
    ALTER TABLE {sessions} CHANGE `session` `session` LONGBLOB DEFAULT NULL [success]
    'all' cache was cleared.                                                [success]

    Finished performing updates

Change Permissions

    ~# chown -Rv www-data:www-data -- /var/www/website/

Go online, clear cache and check your website

    ~# drush vset --exact maintenance_mode 0
    ~# drush cache-clear all