A 5 gal doesnt need a filter... I had a 10 gal without one for a while. Well, technically it was 5 gallons as it was only halfway filled at the time.
If you're going for a natural look you can also add plants to further help with water quality. They look lovely too and bettas seem to enjoy them. If you dont want to mess with aquatic plants and lighting etc, you could just put in some bog plants that root in water. I have some in my 10 gal and it creates a very nice riverbank look My personal favorite is the pothos vine (roots on the left side of the pic). Just clip off a large piece and put it in the tank. You can float it in the water (so long as leaves have access to air) or drape it over the side of the tank. It will sprout roots in no time and with enough plants (a bunch of small clipping will be better than one large one) it will be great for removing ammonia and nitrates from the water in between water changes. It may take a while for them to get used to being grown hydrophonically (sp?) but once they do they really take off.