Typically people use Camtasia Studio to do recording but that costs money.
If you have another computer, here's an idea that you may try:
- get VNC (UltraVNC or any other type)
- set VNC server on comuter that will do the browsing
- set VNC viewer on computer that will do recording
- connect viewer to server - you get a live window into other computer
- use vnc2swf http://www.unixuser.org/~euske/vnc2swf/
to record session to Flash format
Both VNC and vnc2swf are free but you'll need some work to do to make it all work together. Have not tried it myself, so don't know if it'll really work.