Last bit first:
The current rule is that you need a TV licence to watch TV transmissions as they are broadcast (whether that be through terrestrial transmitters, from satellites or via the internet) but you don't need a licence to watch recorded services (such as via the BBC iPlayer). However the Government has announced that the legislation is to be reviewed, with the likelihood that some services which are currently 'licence-free' will in future require a TV licence.
A Smart TV (which is what your post seems to be referring to) requires a router but not a PC. (However, if you've got a PC connected to the same router you can stream content from the PC directly to the TV):
http://www.which.co.u...-tv/what-is-smart-tv/
Chris