We have one at school, and you can connect it directly to either a computer or a TV, so yes you can watch on TV.
The one we have has a 30ft cable, so you'd have to work out where to site it. You can get wireless versions, but they are more expensive and the signal can be a bit dodgy sometimes. Also, unless you use batteries, which would then need changing from time to time, you're going to need a power cable anyway - the model we have has a combined cable which goes through a small hole in the classroom wall.
We haven't had a lot of luck with ours - last year a blue tit laid 13 eggs and the children loved watching the nest - but then a woodpecker bored straight through the box and took ten of the eggs.
The mother continued to sit on the remaining eggs but it soon became obvious they weren't going to hatch - she had presumably been away from them for too long while the raid was going on. She went on sitting devotedly for almost a month though before finally giving up.
This year a nest was started in the box (which our caretaker had practically armour plated) but no eggs were laid.
We will be trying again next year.
We have a live stream to the SouthEast Grid for Learning's birdbox page which you can see here - none of the cameras are live at the moment - we will be settting up again next term ready for the nesting season- but the home page has a lot of useful information about setting up a nest cam and some useful links.