"through the appropriate channels"

Solution 1:

One meaning of Channel is “a way of communicating with people.” The term radio channel came into use to describe a particular frequency of radio waves used for communication and google Ngram shows a growth of the term from the 1920s. The wider use of channel within the term communication channel seems to have come into use from about 1940 onwards. An appropriate channel in this case is a channel of communication appropriate to the matter under consideration. It is used to make formal complaint to those in the management chain charged with dealing with complaints. This would be (as you suggest) the proper channel for complaint (rather than shouting and waving wildly at staff, for example).