Does the structure of water change when a drop of color is added?
No it does not change the structure of water.
Yes the H-bonds between water molecules are broken to accomodate the dye molecules. However, liquid water does not have a set shape or fix molecular arrangement. So what happens is,
The dye molecules simply diffuse out all throught and among the water molecules. The dye molecules become hydrated to some extent, that is, the charged groups and areas on the dye molecule interact with water to form H-bonds or polar bonds which allow the molecule to remain suspended among the water.

Source, General chemistry course work.