Ever try Lactaid milk? It tastes just like milk but with out the lactose. Did you know you can cook with it? Yep. When you need to make something with milk, such as chocolate icing, just substitute Lactaid milk for regular dairy milk. It doesn't change the taste at all, tastes the same as if you had used dairy milk. Here's the recipe for an awesomely delicious chocolate cake by Hershey: https://www.hersheys.com/recipes/recipe-details.aspx?id=184. You can substitute Lactaid milk for the milk that is used in the icing. It tastes the same as if it was made with regular milk.
If you want to use another type of milk, that works too. You can use soymilk, or rice milk. However, soymilk and rice milk have different consistencies (soymilk is very thick, ricemilk is very thin) so it's important to remember that when cooking since it could change the flavor or texture of the food you are makimg.
This webpage can give you tips on how to substitute other ingredients to make your recipes lactose/dairy free. http://www.webmd.com/digestive-disorders/recipe-ideas