All you have to do is let your Alienware PC boot to the black screen, once you see your icon (mouse pointer) then hit the CNTRL key once, even though you do not see it, the password box is now up and once you type in your computer password and hit enter, immediately your icon will disappear. Give it about 3-5 minutes and your Alienware PC will come up. Here's the issue, the alienware x51 and other models believe that there are more then one display attached to your computer and unfortunately it is defaulting a phantom monitor as your main screen, the reason you see black is because you are stuck on the "login" screen but just cant see it. Once you do this you are in!
Here's the thing, once you are in you have to change your display settings in control panel in order to shut off multiple monitors and default your monitor as primary or you wont be able to see anything when you open it as it will be on the "phantom screen". You can do this very easily by opening control panel and hovering over the icon for it on your bottom task bar, then right click and click "move" then move the screen from left to right, it sounds hinky but it works! Once you get the control panel screen in front of you now you can change your display and all will be perfect! This takes a total of 5 minutes and will work!
Credit Alienware discussion boards.