You don't *have* to use a filter, and honestly it would be pretty much impossible to find one suitable for a one-gallon bowl. However, filters do remove particle waste from the water and provide a great place for beneficial bacteria to grow and thus keep up your water quality.
Also, bettas need a temperature of around 78-80 degrees. Unfortunately, a one-gallon glass bowl is so small that temperature fluctuations are pretty much inevitable. In my experience, even the smallest of heaters can't maintain a good temperature in a tank this small and will often overheat the bowl. If I were you, I would look at getting at least a 2.5 tank (a standard 5.5g or even 10g tank would be better and would be about the same price as getting one of those 2.5g kits from Petsmart) and getting a low-flow filter and a heater.
For example, here's an Eclipse 3-gallon kit that would still need a heater for $45
Eclipse System 3 Aquarium by Marineland - Aquariums - Fish - PetSmart
But you could, instead, get a 10g tank for $11 from Petsmart, a cheapo fluorescent strip light from Home Depot for under $10, then buy:
The 20" glass canopy for $8:
Perfecto Glass Aquarium Lighting Canopies
And a filter like this one (the 10i) for $11:
Aquarium Filters: Whisper In-Tank Filters
And there you go...you've spent less money and have a tank that's three times bigger.