I am having trouble with my Alienware X51. When I first set it up I noticed that the lights on the side did not seem to work, however this was a minor thing that I was not that bothered about. I used my TV using a HDMI cable and it worked fine. however, after less than 2 weeks when I turn it on the screen stayed black. I have tried it on both my TV's but depending on which TV I connect it to they just say "Not Connected" or "Check devices power supply, Cable connection, or System" I switched the HDMI cable with my xbox one and that did not help either. My old screen doesn’t seem to have the right bits to connect to it however, so I do not know if it’s because I’m using a TV and not a monitor.

If anyone has any ideas I would much appreciate it!

Could be the graphic card went wonky. Have you tried to hook it up with a DVI or even VGA output if the card allows it? What kind of card is it? Perhaps try it in another PCI slot?
Could be a number of things.

There does not appear to be a slot for VGA. And I do not have a screen with DVI cables which is why I used the HDMI and used it with my TV. I think the video card is a GeForce GTX 555.

I had a similar problem with my X51, you need to plug it into a screen or monitor using the DVI connection (I used a DVI to VGA adaptor to connect it to the tv) just did a windows repair from there and it all worked fine (it had playing BF3 when there was a power cut, not a happy pc).

Do you, or did you ever see the BIOS screen using the HDMI connection?

Yep same exact issue I had. 1st day it got home, I was saddened by the No Device Connected going across my tv screen. So I ran to Staples, got a VGA cord and a DVI converter. Works fine, even though every now and then my whites turn a blueish tent but goes back to normal.