I would say if it doesn't have a filter then you probably need to clean more frequently than that, at least in the early days - especially if the substrate (the gravel or sand in the bottom) has been allowed to dry out at all when you were transferring the tank as that's where all the good bacteria is.
It does sound to me like new tank syndrome (but can happen at any stage of a tank's life cycle if for any reason the good bacteria is killed off).
Basically there is a nitrogen cycle in every tank. Fish excrete ammonia (and goldfish excrete LOTS of it, a filter really is essential for goldies because they are VERY messy fish). When your tank is new or the bacteria aren't there, the ammonia slowly builds up and poisons your fish.
Eventually (assuming your fish hasn't died in the meantime), bacteria accumulate and convert the ammonia into nitrite. Unfortunately, nitrite is also toxic to fish, just not as much.
Then a long time after that, bacteria forms to convert the nitrite into nitrate - also not particularly pleasant for fish, but tolerated in far greater quantities. At this point, you can declare your tank CYCLED. THIS is the point where you can maintain it by weekly water changes.
Unfortunately, if your tank is still in the early stages (and sick fish is likely to indicate this, sorry) then weekly water changes are far too infrequent to keep the toxins at safe levels. Look at doing daily water changes of around 10-30% to keep your fish healthy. When you get your water test kit, you're looking for the following levels:
Ammonia 0 ppm (parts per million)
Nitrite 0 ppm
Nitrate