Create instance of a class with dependencies using Autofac

Have a look at the Composition Root pattern.

You are right, pulling up the dependency resolution only moves the problem to another place. If you continue to move it upwards in your object graph, though, you will reach the entry point of your application. There you will compose your object graph.

Compare that to the Service Locator anti-pattern (using DependencyResolver in client classes in your case) and you will see that Composition Root is a superior solution.

First of all apart from constructor injection you can also use property injection and method injection. However constructor injection is most common and the fastest method so I suggest to stick to it.

The second thing you need to do is to register your MyAwesomeClassin the Autofac container along with its dependency, they have some nice examples right at their home page.

And the last thing - you should not create instances of MyAwesomeClass directly - use Autofac instead. Here is an updated example:

public void AwesomeMethod()
{
    //...
    // AwesomeStuff
    //...

    var GetErDone = DependencyResolver.Current.GetService<MyAwesomeClass>();
}