What is the best technique to put icing on plastic?
Are you sure you want to use icing? The barbie cakes I have seen all have the skirt made from rolled marzipan, not from icing. The bodice can be a normal fabric top or dress (inedible, can be washed after the cake is eaten) or also molded from marzipan. I agree that it isn't as tasty as buttercream icing, but it surely makes a more beautiful skirt. 

This recipe has 196 pictures, you can see different variations of the bodice. 

If you insist on using icing, I would try to smear something sticky onto the doll first, for example honey or syrup in the softball stage, then apply the icing on the sticky layer. I haven't tried this myself (nor have I tried applying icing directly to plastic, so I don't know if it will hold). 

On another note, don't forget to put the barbie's legs in a cut-off paper towel cylinder, so she doesn't get damaged when the cake is cut, and doesn't fall when pieces of the skirt are missing. Also, if you bake the layers in a guggelhupf pan, you don't have to cut a piece out of the middle, which is hard to get right.