You would probably be better off with an actual filtered tank. Even a 3g would be fine for one betta. The filter will allow the bacteria to cycle and filter toxins out of the water.
In a bowl you will not get much bacteria for filtration because there is no water circulation and those bacteria need quite a bit of oxygen. Your betta does not necessarily need oxygen because they are air breathers, but they can also use their gills like other fish, so circulation will help their health.
If you get a filtered tank do not completely change out the filter media because you will dispose of the beneficial bacteria. Rinse the filter media in tank water when you do a water change and put it back in the filter.
You do definately need water conditioner unless you have well water, because you need to remove the chlorine. This can also be accomplished by aging the water (filling a bowl and letting it sit for 24-48 hrs so the chlorine can dissipate). Never add water directly from the tap without dechlorinator (unless you do know for sure you have well water, in which case you don't have chlorine).
Always make sure the water you add is the same temperature as the tank, or else you can temperature shock your fish.
Temperature btw, should be about 75 or a bit higher, since bettas are tropical fish. They don't do well in lower temperatures for long periods. They also will not be active in low temperatures since they're cold.
As far as your original comment about him laying on the bottom of the bowl, this is pretty common for bored bettas. It does not mean he is ill. Even in a filtered tank a betta will sleep against the intake for the filter and appear to be dead. Bettas need to be mentally stimulated, and you can do this by putting a mirror against the tank for a few minutes a day, which will cause him to attack his reflection and exercise/stimulate him a bit.
Male bettas are not very active fish unless something gets them riled up.
You should be feeding him betta pellets mainly. He only needs about 6 a day (do not feed him until he stops eating and is fat, it's not good for their health and can foul the water in your tank). You can feed a few bloodworms sometimes for a treat, which will give them some good protein.
Does he ever make a bubble nest on top of the tank? This is a sign of them being pretty content. If he never makes a bubble nest he may be stressed or unhappy (though I have noticed some bettas don't make bubble nests, most do.).