routes.php

<?php

Route::get('/tweet', ['as' => 'tweet', 'uses' => 'TweetController@index']);

TweetController.php

<?php

class TweetController extends \BaseController
{
    protected $tweet;
    
    public function __construct($tweet)
    {
        /* How do I access $app['tweet'] in this controller? 
         * without using App::make
         * 
         * I want to inject through the constructor and still be able to use route the normal way
         */
        $this->tweet = $tweet;
    }
    
    public function index ()
    {
        $this->tweet->send('Here is my message');
    }
}

TweetServiceProvider.php

<?php

class TweetServiceProvider extends ServiceProvider
{
    public function register()
    {
        $app = $this->app;

        $app['tweet'] = $app->share(function ($app) {

            $storage_provider = null;
            $request_provider = null;

            $config = array(
                'key'        => $app['config']['tweet.key'],
                'secret'     => $app['config']['tweet.secret'],
                'token'        => isset($app['config']['tweet.token']) ? $app['config']['tweet.token'] : null,
                'token_secret' => isset($app['config']['tweet.token_secret']) ? $app['config']['tweet.token_secret'] : null,
                'auth_method'  => isset($app['config']['tweet.auth_method']) ? $app['config']['tweet.auth_method'] : null,
            );

            return new Tweet($config);
        });

    }

    public function boot()
    {
    }
 }

Inside boot():

$this->app->bind('TweetInterface', 'Tweet');

Then typehint it within your constructor:

public function __constructor(TweetInterface $tweet){}

When it's resolved from the IoC, it'll find the binding of the interface and inject Tweet as the dependency.
Bind also accepts a closure, so you could return App::make('Tweet') within the binding.

Create and implement the interface on the classes too!

Thanks @acairns I think I should have been more clear with my question, this is a third party package and I wouldn't want to hack the package. Would App::alias('tweet', 'Tweet') work? I think there should be some docs around creating third party package creation.

Because I think this is how it should have been done:

<?php

class TweetServiceProvider extends ServiceProvider
{
    public function register()
    {
        $app = $this->app;

        $app['Tweet'] = $app->share(function ($app) {

            $storage_provider = null;
            $request_provider = null;

            $config = array(
                'key'        => $app['config']['tweet.key'],
                'secret'     => $app['config']['tweet.secret'],
                'token'        => isset($app['config']['tweet.token']) ? $app['config']['tweet.token'] : null,
                'token_secret' => isset($app['config']['tweet.token_secret']) ? $app['config']['tweet.token_secret'] : null,
                'auth_method'  => isset($app['config']['tweet.auth_method']) ? $app['config']['tweet.auth_method'] : null,
            );

            return new Tweet($config);
        });

        // This should always be the way, this way I can get via Alias or by Full Class Name to use Class Automatic Resolution
        $app->alias('Tweet', 'tweet');
    }

    public function boot()
    {
    }
 }