Whenever there is fish food or fish waste in water it produces ammonia. Over time (usually 2-4 weeks) the filter develops a colony of beneficial bacteria that eat ammonia and convert it into nitrite. A few weeks after that and more good bacteria develop and convert the nitrite into nitrate. When the filter has all these beneficial bacteria working properly the filter is considered cycled.
Right now I would say you are about half way thru the cycling process and have ammonia and nitrite in the water. These are both bad for the fish and will be the most likely cause of the tail and fins splitting and breaking down.
You need to do a 75% water change and gravel clean each day for the next few weeks. This will dilute the ammonia and nitrite in the water and make it safer for the fish.
Make sure any new water going into the aquarium is free of chlorine/ chloramine. Tap Water filters do not always remove chlorine/ chloramine so make sure you use a dechlorinator in any new water before it is added to the tank. The best way to do this is to get 2 clean buckets that have never been used for soaps or chemicals and write "FISH ONLY" on them. Use the buckets for the fish and nothing else. Fill one bucket with tap water and add some dechlorinator, then aerate it for at least 30 minutes (preferably longer) before you use it. Use the other bucket to put the water you drain out of the tank, then tip the old tank water on the garden. Because you are going to need to do water changes each day for a few weeks I would suggest making up the bucket of water each day, use it for the water change then make up another bucket immediately and allow it to sit near the tank until you use it for the next water change tomorrow.
If you don't have a gravel cleaner, visit the local petshop and get one similar to the one in the following link. You only need a small basic gravel cleaner, nothing fancy.
http://www.about-goldfish.com/aquarium-cleaning.html
You could also invest in an ammonia test kit, a nitrite test kit and a pH test kit. Buy liquid test kits not paper strip test kits. The liquid test kits are more accurate. If you can't afford the test kits, take a glass full of tank water to a petshop and have them test the water for
ammonia, nitrite & pH, and get them to write the results down on paper for you. Then post the results here.
Do not bother adding medications/ chemicals to the tank at this stage, just do big water changes and gravel cleans and see how he looks in 2 days. If it continues to get worse after a couple of water changes then you will need some medication. But try water changes first.
Right now you need to do a water change first and foremost. You don't have to do the gravel now just do a big water change. Then visit the petshop when you can and get tests done and buy a gravel cleaner. Take a water sample for testing before you do the water change.
Live plants would help and if you can find some Water Sprite (Ceratopteris species), that will float in the tank and help use some of the ammonia and nitrite.
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Do you live in a warm climate or is the room the fish lives in just naturally warm? I ask because you don't have a heater. Bettas are tropical fish and need warmth. Your current temperature is fine but if it get cold you will need to invest in a heater for the aquarium.
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Only feed the fish once every second day until the filters have established. Then you can feed more often. The more food you put into the tank, the more waste and ammonia that will be produced.