We have Netflix and Hulu accounts. For device-to-tv, we stream through a PS3 in one room and a Roku in another.
Modern streaming devices work with wifi. Some can also be wired if you choose to run lines. In either case, they don't go through your computer, so you don't have to have it on to stream things.
Using your computer to do other things while someone is streaming doesn't normally impact anything unless you have a slow connection. If you have high-speed internet there should be no problems.
Added: After reading the other replies, I wanted to expand on Gamma's comments about how streaming works to clear things up -
1 - "ANYTHING that you stream is like pouring money out of your pocket into the internet providers pocket."
This is a Maybe, not an absolute. We don't pay anything extra to our ISP for streaming.
Streaming uses quite a bit of data so if you have a limited access internet plan with a data cap, or if you use your cell phone service for internet, then you can eat up your allowance quickly and end up paying more in add-on data and fees. If you're going to stream it is best to have a high speed connection with unlimited data.
2 - "It's flat out cheaper to have a provider for your TV."
This could vary by area, but I can tell you that our *combined* cost for our internet, Netflix and Hulu is significantly less expensive than the cheapest low-level cable and satellite packages offered in our area. We ditched the cable years ago and the cost savings is enormous.
3 - "...you also have to pay fees to the companies you stream from."
Yes, sometimes. Netflix base price is $8.99 if you're not grandfathered. Hulu has both Free service and paid service. Roku has a lot of free channels, as well as optional pay channels.
4 - "Like if you want to watch NCIS you have to pay a membership fee each month to the major network to get access."
It is true that some networks do this but you don't have to pay if you're patient. Past seasons for most things show up in full for no added cost on Hulu or Netflix. Then you can binge watch at your leisure. Using the NCIS example, Netflix has seasons 1-12 available to stream. The current Season 13 will probably be available in July.