How to make a video where an actor writes facing the camera?
You need a few simple things to do this:


a camera
a large piece of glass or plexiglass
dry erase markers
editing software capable of basic transform controls


Record your actor writing on the glass.  To the camera, the writing will appear backwards.  Use your editing software to flip the image horizontally.  The writing will now appear normal*.  Most editing software is capable of this operation.  If there's not a "flip horizontal" command, try setting the horizontal scale to -100%.

*Pro tip:  Make sure your actor's clothes don't have any writing on them, or it will appear backwards in the final result.