The main brook char fishery in Canterbury (and New Zealand) is Lake Emily in the Ashburton group. It is fly fishing only water, for obvious reasons the char take the same type of flies as the other trout in the lakes. Brookies are schooling fish, which often stay stationary rather then cruise the edges. So slowly work the edge of the lake, if you spot one there is likely to be more around. They are hard to spook, anything short of being caught would often see them return to their lay after a few minutes. It might take a few trips to get onto them. They are not exactly plentiful.
Due to the fragile state of the fishery, I advise not to keep any brookies and release any which is likely to survive. These also Brown trout in Emily, whack them on the head and throw them into the tussock, the brookies do not need such competition.