Drupal - "Drupal calls should be avoided in classes, use dependency injection instead"

Take the BlockLibraryController class as example; it extends the same class as your controller.

You define:

• A static and public create() method that gets the values from the dependency container, and creates a new object of your class
• A class constructor that saves the values passed from the previous method in object properties
• A set of object properties to save the values passed in the class constructor

In your case, the code would be similar to the following one.

class MyModuleController extends ControllerBase {
  /**
   * The path alias manager.
   *
   * @var \Drupal\Core\Path\AliasManagerInterface
   */
  protected aliasManager;

  /**
   * Constructs a MyModuleController object.
   *
   * @param \Drupal\Core\Path\AliasManagerInterface $alias_manager
   *   The path alias manager.
   */
  public function __construct(AliasManagerInterface $alias_manager) {
    $this->aliasManager = $alias_manager;
  }

  /**
   * {@inheritdoc}
   */
  public static function create(ContainerInterface $container) {
    return new static(
      $container->get('path.alias_manager')
    );
  }

  /**
   * {@inheritdoc}
   */
  public function getUserContent() {
    $alias = $this->aliasManager->getPathByAlias($_POST['url']);
    // Omissis.
  }

}

Don't forget to put use \Drupal\Core\Path\AliasManagerInterface; on the top of the file containing the code you are showing.

As side note, the code you use to render the view is wrong: You don't need to use drupal_render() because views_embed_view() already returns a renderable array.
Then, the render array you are returning is probably not giving the output you expect. #name is probably not going to be used from Drupal, and #markup filters the markup you are passing to it, as described on Render API overview.

• #markup: Specifies that the array provides HTML markup directly. Unless the markup is very simple, such as an explanation in a paragraph tag, it is normally preferable to use #theme or #type instead, so that the theme can customize the markup. Note that the value is passed through \Drupal\Component\Utility\Xss::filterAdmin(), which strips known XSS vectors while allowing a permissive list of HTML tags that are not XSS vectors. (I.e, <script> and <style> are not allowed.) See \Drupal\Component\Utility\Xss::$adminTags for the list of tags that will be allowed. If your markup needs any of the tags that are not in this whitelist, then you can implement a theme hook and template file and/or an asset library. Alternatively, you can use the render array key #allowed_tags to alter which tags are filtered.

• #allowed_tags: If #markup is supplied this can be used to change which tags are using to filter the markup. The value should be an array of tags that Xss::filter() would accept. If #plain_text is set this value is ignored.