What is the liquid that forms after chicken goes through the freeze-thaw cycle, and how can I extract it more efficiently?
When you freeze meat the water in the cells crystallizes into ice, and in some case ruptures the cell walls. When you thaw the chicken again, the contents of the ruptured cells' are free. These are mostly the water, and the cells proteins.

Additionally, processed meats often have some water and additives added to the at the factory. 

The "juice" you are seeing is probably a combination of these things.