You do not need a filter if you don't plan on cycling the tank. PLEASE look up "aquarium cycle" and read the post here. Even if you do not plan on cycling (ie the buildup of benificial bacteria to process harmful fish waste) the tank, this knowledge will help you to understand water quality and how it affects your fish.
One thing you ABSOLUTELY need for a betta is a heater. A 25 watt adjustable visi-therm stealth heater is my favorite and really the only heater I would trust in such a small tank. Other cheaper brands could easily heat up too much and kill your fish, while those small unadjustable pad ones will cause the temperature to fluctuate with room temperature. Bettas are true tropical fish and need a temperature from 78-82. I have found out that mine do best at 80 F.
If you choose to not cycle the tank, you will need to do 100% water changes every other day because that is a very small environment (I would actually really recommend getting at least a 2.5 gallon instead). You will need to match the temperature exactly and use dechlorinated water from the same source each time. This leads to the fact that you will need a good dechlorinator. If the tank is uncycled you don't need a filter which will likely do more harm than good in a 1 gallon (bettas don't do well in current).
You can feed your fish betta pellets, frozen/freeze-dried bloodworms, brine shrimp, and daphnia. I also use bettamin flakes which most fish seem to love. Don't get tropical fish flakes as these contain too much vegetable matter which can lead to health problems and constipation. Remember to rehydrate freeze-dried food before feeding and thaw frozen food. A varied diet is always healthier for your fish, but you do not need to overfeed. If for some reason you end up with uneaten food, you can suck it out with a turkey baster so that it does not foul the water.
I also highly recommend that you read the betta care guide under "betta fish" in the "betta fish care" section.
