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You need to create a module, and define a menu path in it using hook_menu(), and the page callback for that script will be your upload script.

function hook_menu()
{
  $menu['mymodule/upload'] = array
  (
    'title' => 'upload',
    'page callback' => 'mymodule_upload_callback',
    'access callback' => TRUE
  );
  return $menu;
  //); oops - but not a big enough edit to be accepted
}

Note that you'll need to name your hook in line with Drupal hook naming conventions: https://www.drupal.org/docs/7/creating-custom-modules/understanding-the-hook-system-for-drupal-modules

Then your upload callback:

function mymodule_upload_callback()
{
  // Handle your upload here. You will have
  // access to public:// and the rest of the Drupal API.

  // call drupal_exit at the end() to prevent your
  // script from going past this point.
  drupal_exit();
}