How am I supposed to upgrade a site to latest core version?
From Drupal documentation :


  Strictly speaking, 'updating' refers to taking your site from one 'minor' version to another, as for example, from Drupal 7 version d7.1, to d7.2; or from Drupal 8 version d8.0.0 to d8.1.3, etc.


Which means upgrade can be done only to minor version.

If you want to change the whole version for example moving from Drupal 7 to Drupal 8 you need to migrate your site, and this normally involved a data migration and restructuring, Also changing and reviewing your custom modules code.

See Upgrading from Drupal 6 or 7 to Drupal 8