One thing to remember is that ancient china (and arguably modern china) is not one monolith, but instead a series of empires.

These empires had different leaders, and were multiethnic, but the bureaucracy remained largely the same.

The "weak China" in the 19th century was:

1. A China ruled by foreigners. The Qing dynasty was ruled by the Manchu, a nomadic peoples from NE china. The common Chinese (Han) detested Manchu rule. It was declining in the 19th century, and many Han were desirous for the Qing dynasty to end.

2. A China ruled by bureaucrats. China had a fixed amount of bureaucrats (Mandarins) that helped run China. They were like governors. Interestingly enough they never increased the number of Mandarins even when the population grew. So the government lacked the ability to manage efficiently and did not scale with the population

So you have a foreigner led dynasty that was hated by the commoner which was already in decline, as well as a growth in population without a growth in government bureaucracy.

There are other factors, but these two would be perhaps the foundational factors on which the others hinge.