How can I record video with a Canon 1Dx DSLR which will feed straight into Adobe Premiere or Final Cut?
Any DSLR that can provide a live video signal (such as via HDMI) can feed that signal to a recording device (such as Atmos SHOGUN) or monitoring device (such as a TV monitor).  In order for your computer to handle live video, you need a live video input port, typically found on cards such as the Blackmagic Design Deck Link card.  BlackMagic's Media Express software can record and/or monitor live video.

I would be surprised if Adobe Premiere Pro (or other NLEs) can handle live video as an editable stream.  But MediaExpress can record with a confidence monitor, and then Premiere Pro can edit the resulting file.