You are supposed to keep your compost pile watered, as well as your gardens, to keep the microbes happy and to make compost quickly. The runoff is the compost tea. Although you can just make compost tea without the big pile. Go here to read more: http://www.finegardening.com/how-to/articles/brewing-compost-tea.aspx
Thanks for the answers, but I'm still hoping for more ideas.
Put a few handsful of finished compost in a bucket and fill it with water. Let sit for a few hours, then use the liquid to water your plants.