I have a 42" vizio television.  I plan to use this as my monitor.  Does the Aurora have a HDMI output?  If not, how would I go about connecting the PC to the television?

My Aurora came with some adaptors in the box, one being an DVI to HDMI adaptor. Now this is only for Video, for the audio you'd have to use either your computer speakers, or run an audio out from the PC to the TV, or to a receiver.

If your pc came with an alienware monitor and depending on your video card...   there should be a HDMI cable.

you should be able to connect that from your video card to your lcd tv and get full hd signal

What video card? If it's a 5000 series ati than yes. You will get sound off of your video card as well through the hdmi.