For slightly different needs, I too would like to figure out the best way to do Skype on TV (for elderly grandparents). A webcam (thinking Logitech c920) connected to a device ought to do the trick, but what that device should be I don't know. A PC Stick with an Atom? An Inspiron 11 3000? (trying to keep the device down to < $300). Thoughts appreciated
No need to reinvent the wheel, especially for people that won't be able to do their own tech support. Each extra component you're working with can lead to extra problems..
There are lots of devices built in the past few years that do this perfectly. Do you specifically want it set up on a TV size screen, or would a tablet size work?
Tablet size (benefit: portable, take with you to any room in the house)
Google makes the Nest Hub Max
Amazon makes the Echo Show
TV size (benefit: large screen)
Facebook makes the Portal TV (also available in tablet size)