Have seen this error happen do some files not updating properly. There are some files that Hearthstone uses that are not normally removed during a normal uninstall. Delete these files here. Then uninstall Hearthstone and reinstall it.
I am not sure specifically what you are asking. If you are looking to make sure all process for Hearthstone and the App are not running use Activity Monitor to quit all Agent, Battle.net, and Hearthstone processes.
If you are asking how to do the steps I linked on a Mac, the pages I linked have a Mac tab. Click on that tab to see the Mac instructions.