UPDATED! 11 Nov 15These instructions should allow you to install Blizzard Updater and Hearthstone on Linux using Wine. From what I understand these instructions will allow some of the other Blizzard games to be run on Linux as well.

First install wine 1.7.
In a terminal type:

sudo add-apt-repository ppa:ubuntu-wine/ppa

and then:

sudo apt-get update; sudo apt-get install wine1.7 winetricks

Now you need to add your library exceptions. The winedb website says to add the wininet lib in winecfg but if you simply install wininet from winetricks it automatically adds the right setting so in the terminal again:

winetricks wininet allfonts

Allfonts installs several windows fonts. It may not be absolutely necessary but it’s quick and easy to install and you really can’t go wrong doing it.

Finally run winecfg from the terminal or find “Configure Wine” in your desktop menu. Go to the “libraries” tab. You should see wininet( native,builtin) already listed there. Leave it be.

In the field labeled “New override for library” type in:
dbghelp
click add
click Yes on the dialogue that pops up.

In the field labeled “New override for library” type in:
msvcp100
click add
click ok

You’re all setup. Download and install the client from http://us.battle.net/hearthstone/en/