You have a couple approaches to change the "user" Drupal path to "my-url":
First, you imitate the way Drupal defines the "user" path on user core module. In that way your "my-url" page will have same behavior than the original one.
Then, to complete the work you will need to redirect all "user" requests to your "my-url" path. To do that you can use custom code by using the hook_menu_site_status_alter() hook or you could use the redirect module.
Here's a quick implementation with custom code:
/**
* Implements hook_menu().
*/
function yourmodule_menu() {
// Registration and login pages.
$items['my-url'] = array(
'title' => 'User account',
'title callback' => 'user_menu_title',
'page callback' => 'user_page',
'access callback' => TRUE,
'file' => 'user.pages.inc',
'weight' => -10,
'menu_name' => 'user-menu',
'file path' => drupal_get_path('module', 'user'),
);
return $items;
}
/**
* Implements hook_menu_site_status_alter().
*/
function yourmodule_menu_site_status_alter(&$menu_site_status, $path) {
if ($path == 'user') {
drupal_goto('my-url');
}
}
Note: the redirect module is not integrated with features module. If you want to propagate automatically this change to all your environments I suggest you to use custom code.