You would need to have a live screen capture program that would let you record anything playing directly off your computer screen. There are free programs that would enable you to manage this, like 'Format Factory' program http://www.formatoz.com/download.html. It is a free download that I have used this in the past to record live classroom computer instruction.
Another option is that there are free websites (like www.keepvid.com) that will download and change the video file format to a format that you could use on your movie editing software. If you google 'free download youtube,' it will provide you with many different options that could work on your computer.
Further Reading