How to color Jiaozi (chinese dumplings)?
You could add chopped fresh herbs to the dough. I'm not familiar with Jiaozi to know how much it would affect the texture, but its common in pasta dough. If you chop it fine enough, it should bleed quite a bit of green color into the dough.

I don't know any particularly traditional food coloring methods, but I do know that tea is often used in place of water to change the color and taste of certain dishes (e.g. tea eggs). However, I doubt that is a flavor you really want to introduce into the dumpling dough.