Hi,
It would really help us a lot if you told us how you are keeping the betta right now. What size tank is he in? What temp? Are there other fish with him? (I assume not) Do you have plants in the tank? What exactly are you feeding him? Any details will help us help you.
It sounds like you are not using filtration, and I don't think that's a good idea. If he's alone in a small tank, you should use at least a box or foam filter. The wastes produced by the fish need to be "processed" by the filter to avoid a buildup in the water. Also the agitation produced by the filter, however slight, will promote gas exchange (with the air) in the system which will keep it healthy.
While it's true that wild bettas can live in puddles and such, it isn't the ideal environment, even for them. Add to that that the bettas we get at the pet stores are highly bred and modified forms of their wild counterparts, and are not as resistant to adverse water conditions as the wild ones. Store bought bettas should be kept in clean, well aerated and filtered tanks, just like any other fish you would buy.
I do not recommend using substances on fish unless you know it is absolutely necessary. I have never used Stress Coat on any of my fish, and yet they lived and were healthy. I'm not even exactly sure what is actually supposed to do, beyond the manufacturer's claim of conditioning the skin and slime coat of the fish.
West871 is right that Chlorine is bad for fish. In it's pure form as a gas, it's deadly to people too. It's used to keep bacteria and other harmful micro-organisms from living in our tap water. The reduced levels of Chlorine present in our tap waters obviously do not harm us, but because fish have to live in it, and breathe it through their gills, it affects them quickly and badly.
There is something even worse, called Chloramine, which is formed by Chlorine joining with Ammonia (usually present in trace amounts in aquariums.) It is also often added by many cities to the tap water. This is an even more deadly poison for the fish, and unlike Chlorine will not "evaporate" if left standing.
But that does not mean you can't use tap water. In fact, as you found out, you have to use tap water, at least in part.
There are 2 ways of dealing with Chlorine. One is to fill a clean receptacle with tap water and let it sit for 2 days to let the Chlorine evaporate away. If your city treats the water with Cloramine though, this will not be enough. (You can find out by calling your city's water department)
The other is to use de-Chlorinating products obtained from the pet store. By adding the recommended amount of de-Chlorinator to the tap water, you can use the water right away instead of having to wait. (make sure it's the same temperature as the water in your aquarium though) It would be a good idea to make sure that the bottle also mentions that it treats for Chloramine as well, to be extra safe.
It seems to me that you are relatively new to keeping fish? If so, the biggest recommendations I can make is to do as much research as you can to find out how to keep your fish happy and healthy. I can see you have already started to do this.
The second biggest tip I can give you is not to keep changing things for the fish. While changing 10% of the water weekly is recommended, changing all the water every couple of days can be deadly. Other over-fussing issues such as adding substances (Stress Coat, medecines, certain water conditioners) or constantly redocorating the tank can also harm your fish. Keep your fish in the water that it's in for now. If there was Chlorine present, the harm has already been done, and you'll only be stressing the fish more by doing another water change. You can add a few drops of de-Chlorinator directly to the water to remove Chloramine if you feel it's necessary. Use only a quarter of the recommended dose if you are adding it directly to the tank. He's still a new fish, and needs to adapt to his surroundings, so let him have a clean and peaceful tank to explore.
Hope I haven't overloaded you here.
The main thing is to A) provide to his needs, and B) relax, and enjoy watching him.
Take care!
Chris