The easiest way to stream legally would be subscribe to the NHL's NHL.tv service, and then because you wish to avoid their blackout rules of like 48 hours in the local market, you would use a VPN to geospoof your location to be someplace else, like Australia with whom no one has broadcasting rights from the NHL.
I run mine through a Chromecast. I just either cast it to the Chromecast attached to the TV via the Chrome browser or via the NHL app on my phone in conjunction with the Chromecast app. My setup is older though and the newest Chromecast works more like a Roku where you run the app directly on the device plugged into your TV and use a remote it comes with or presumably an app on your phone serves as the remote.
If you were to use this type of setup, you would need to connect your Chromecast to the VPN rather than just your home network. You basically create a virtual router/wifi network in addition to your normal network, but that one is running through the VPN from whichever country. So everything involved gets run through that. You can get a service that basically just runs an extension in your browser to do all of this instead of you setting up new networks and whatnot.
As for how much this would run, NHL.tv is $99USD this season, Chromecast is $35 or $65 if you want it for 4k HD which not a lot runs at yet, and VPNs look to be under $5 a month.