The auxiliary verb is usually written in hiragana.
In additon to the explanation, いく/くる(きた) can also express continuous action/movement in the time line. いく is for the future, and くる(きた) is for the past. Plus, ~てくる usually means "to go somewhere to do something and come back".
e.g.
買い物してくる: to go shopping and come back
1) テレビを見ていく:
テレビを見てから行く "to watch TV and go somewhere"
(これからもずっと)テレビを見続ける "to continue to watch TV"
2) テレビを見てくる:
テレビを見てから来る "to watch TV and come here"
テレビを見に行って帰ってくる "to go somewhere to watch TV and come back"
4) テレビを見てきた:
テレビを見てから来た "watched TV and came here"
テレビを見に行って帰ってきた "went somewhere to watch TV and have come back"
(いままでずっと)テレビを見続けてきた "to have continued to watch TV"
#3 テレビを見て来て is a command.
Go (somewhere) to watch TV and come back.