You can't separate the two. Building up the bacteria is establishing the nitrogen (ammonia) cycle. You are concerned with building up beneficial bacteria while keeping the ammonia and nitrite low enough to support fish life.
There was an old assumption that the higher the ammonia level, the faster the rate of growth. But this isn't so; if there's a measurable ammonia level then the bacteria are already eating as much ammonia as they can and there is some left over. The assumption is akin to assuming that you can eat more apples if there are 10 in the fruit bowl than if there is only one*. If there's always at least one apple spare, you can eat them just as much as if there's always a surplus of 10. The assumption, therefore, is false, and the advice (which is rather old hat nowadays) to avoid large water changes which is based on it is also false, and indeed dangerous as the elevated ammonia and nitrite can, and regularly do, kill fish.
The level of nitrate isn't relevant to whether a tank is cycled or not, but whether it's receiving enough water changes. Indeed, if you don't do sufficient water changes during the cycle you will end up with a very high level of nitrate.
*Actually, humans probably will eat more apples if they can see more spare. Bacteria, however, do not have the same psychological hangups as us about eating the last one.