I’ll answer your question right now without beating around this bush, what you are asking is simply impossible. In fact I will take it one step further, not only is your request not possible using your preexisting TV there is no such consumer TV that you can purchase as of now that will just run Linux. Furthermore there is not a single Linux distro as of now designed to run internally inside of TVs (there are distros for set top boxes, not TVs).
Is it possible in theory?
Possibly but until a single confirmed case of a successful Linux installation on a TV is found I will consider this impossible (furthermore the chances that the successful Linux installation occurs on your specific model of TV is slim).
The requirements to make replacement Linux firmware for a TV would be
At that point just make your own smart TV using a commercial display