0ppm ammonia out of the tap is perfect, that's a good start in addressing your problem. The fact that your tank water has ammonia in just screams inefficient filtration and/or overfeeding and/or lack of proper maintainance, or maybe all of those things combined.
I can't really see it being your bio load as you've only got one fish, unless it's a very messy fish and you regularly overfeed this one fish, in which case that could be your issue right there.
When you do your water changes, and I agree with duanes by the way, that your water change schedule could be increased a bit, is there a remote possibility that untreated water is getting flushed through your filtration because that would mean you are potentially wiping out your BB in which case your BB will never be able to multiply to the levels needed to maintain the nitrogen cycle.
That would lead to constant showings of ammonia, which is what you're experiencing.