How to Upgrade Drupal Core manually
Unfortunately, Drupal didn't allow users to automatically upgrade Drupal core. If you plan to use latest version you should upgrade it manually. Up to date version is mandatory for security purpose. Note that your PHP version must be 5.5.9 or higher otherwise you cannot use Drupal 8, see the system requirements for further information. Contact your hosting provider to configure your PHP version.
Upgrade Drupal core
First of all, login to drupal dashboard. To checkout update navigate to Reports - > Available updates -> Check manually. There is where, update manager for drupal core, themes and modules. Once you click on checkout manually link it will find for available latest version. As a result it will show outdated results. If you're using creepy version, download latest version of Drupal from official drupal site. Now explore the file you downloaded using archive viewer and delete /sites/ folder from that. Connect your host via FTP or open web explorer/ file manager on your hosting. Backup your whole site. I highly recommend you to back up your files even if you want made a small tweaks.
Upload the compressed file to server. If you've access to Cpanel extract compressed file via file manager otherwise extract the archive to your computer and upload using any FTP client. Replace old flies with newly uploaded files. Then upload site folder and .htaccess, robots.txt from your backup file. Note that, sites folder has some sensitive files predominantly your site's configuration ( drupal stored your configuration in settings.php file), themes, modules, Images files. Once you finished, run the update manager. Your drupal core will be up to date. After that run the update script and complete required option finally disable maintenance mode.