Have you used any meds, though i wouldnt expect a high nitrate reading.
What live plants do you have in the tank.
I would change the black carbon if its leeching something into the tank as it wants changing.
What substrate do you use.
Sorry you replyed to the black carbon.
Sounds like your filters are not working properly, as you are doing everything right.
Racked my brain over this, high nitrates with a high nitrite reading says to me like the filters are not coping as though it can't keep up.
Your tap water stats you say are fine.
You are doing gravel vacs and water changes.
Fish tank is not overstocked.
Not over feeding.
Cloudy water can be a bacterial bloom but that not it as you have a nitrate reading.
Also cloudy tanks lack of maintainence but you do them weekly.
Cloudy tank can mean filter not working right or its not coping.
http/www.aquahobby.com/articles/e_waterchange.php
Just want to make sure you don't use an undergravel filter as well do you.