How can I record demos of my gameplay?
How does one capture gameplay video on PC?
I am currently playing Skyrim, how can I capture the video of my character?
Do I need new hardware to capture my gameplay video?
You don't need new hardware, just software. There are plenty of free screen capturing softwares on the web. Try sorting through here : http://techie-buzz.com/softwares/top-free-screen-recorders-to-record-your-screen-activities.html . Should be a good start.