You can only legally watch Chicago Blackhawks games online through a subscription to Gamecenter Live from NHL.com. The subscription service also has a dedicated mobile app for streaming live games.
You need a subscription to Gamecenter Live from the NHL's own website or mobile apps to watch any live Blackhawks game online. The subscription allows users to stream any out-of-market game by any NHL team live. The league bases the market status of individual games on the home location of the user, so if you live in the Chicago designated media area, you can only stream the Blackhawks' away games using this service.