I own a Rheem Criterion II furnace in my home and I was wondering how to clean the furnace by myself to try and save me money.

That would be best left to a professional. They can also check for the proper operation of your furnace at the same time.

About the only thing you can really do is wipe off the dust unless you feel like taking the unit apart to vac the dirty areas and pull the blower to clean the cage etc. I do not recommend you do this unless you have the necessary mechanical skills and knowledge.

What you might do is call to have the work done then watch the serviceman do the cleaning. This way when it needs to be cleaned again you will know what it entails.