This is called Ongoing Harassment by Blizzard. There are a few things you need to do so they can best help you with it.
Read this post on it. It will explain how they view harassment and what steps need to be done.
Putting them on IGNORE and not talking to them at all is exactly the right thing! Then you document the name-server, chat channel, and time stamp (include time zone) of each time they contact you on another account. Submit that list via ticket to Blizzard using the Ongoing Harassment ticket option.
Do not ever talk to them or Blizzard considers it mutual contact.
They should not be able to use an alt on the same account to contact you. They should be forced to use a new account. That catches up with people.