Never buy alienware unless you want the look of the case more than you want performance.
Personally, If you're not up to building it yourself I can guarantee you will have a hard time finding a small MATX case or even smaller gaming computer. The alienware desktop is their take on a steam machine. Something small like a gaming console yet able to do anything a normal gaming desktop PC can do. Don't waste the money on alienware. I can assure you that you will be disappointed a year or two down the line and wonder why you spent so much cash on such little performance. I can tell you right now that you are paying for a designer case. Its basically a woman's Gucci hand bag or something like that crap.
As for the OS, I would recommend Linux but unless you actually know how to use Linux/Ubuntu it can be very confusing and lots of games are not supported for Linux YET. They are making great progress though for a free operating system.
Your realistic budget would most likely be anywhere from 400-600 USD since you have to factor in an operating system and a monitor. You're generally looking to spend ~$100 on an OS and ~$100 on a monitor of your choice. Instantly slashes your price range almost in half.
My ultimate recommendation for such a low budget - Buy your parts one step at a time. If you can, every time you get a paycheck just take anywhere from 100-200 USD out of that to go buy a part for your PC. It will take a bit longer than usual. I'd say that process would take maybe 2 months to get your full PC up and running and although people might not agree with this, I would suggest you either save some more $$$ cha-ching cha-ching or buy the parts one at a time.
If you want some guys to build up a PC parts list for ya I am sure any of us can throw you some decent suggestions.
And now that I think about it - Gigabyte just released their new Brix pro with Intel Iris graphics. You can generally pick up a bare bones unit for ~700.
http://www.newegg.com/Product/Product.aspx?Item=N82E16856164012
I remember making a thread and posting a video of it playing Bf4 on maximum settings at 720p with a decent 60 FPS. Even playing 4k video.
Intel Iris pro graphics is a pretty large step for intels gpu lineup.