Overall, what weaknesses led to the collapse of the Qing dynasty?

There were at least two main weaknesses that led to the
collapse of the Qing (or Ch'ing) Dynasty ( href="http://www.artsmia.org/art-of-asia/history/dynasty-ching.cfm">1644-1912). 
The government was incompetent and corrupt and its military was too weak.  These two
interrelated problems led to its fall.

Over the centuries,
the population of China grew rapidly, putting more pressure on the government to police
and care for the population.  However, the government did not expand or modernize. 
Instead, it kept to its old ways which had been more adapted to a smaller population. 
Thus, as the population grew and social and economic problems resulted, the government
was unable to deal with those problems.

At the same time,
the weak military allowed foreign countries to dominate China wherever they met.  The
Opium Wars were the clearest sign of this.  This domination by the West disenchanted
many Chinese.

Overall, then, the government was too
corrupt, ineffective, and weak to police, protect, and care for its growing population. 
This led to widespread dissatisfaction and the eventual collapse of the
dynasty.