When you launch Heroes of the Storm directly, it would just take you to Battle.Net. The same is the case with Diablo 3. Adding the parameter -launch to D3 circumvents Battle.Net though and you can log in and use the Steam Overlay by adding D3 with this command line parameter as a Non Steam Game.
However, this does not seem to work with Heroes of the Storm. Is there any way to launch Heroes of the Storm from Steam with the Overlay enabled for streaming?