The verb 「みる」("look"/"see"/"watch") is one case of a word which can be written with one commonly-used general-purpose kanji, and sometimes with other rarer, more specific kanji.
The general-purpose kanji writing is 「見る」. Any time you use 「みる」, you can be confident that you can write it 「見る」 and it will be correct (as long as it's a verb which means anything like look/see/watch). Here's a Japanese dictionary entry with a variety of meanings for 「見る」.
As the same dictionary entry says, 「観る」 is a way of writing 「みる」 which can be used when it has specific meanings:
- Looking at scenery, as a tourist does, and
- Watching something for entertainment, such as TV, as in your example.
In these cases, 「見る」 is also OK, so you don't have to use 「観る」, but if you do, it makes the meaning more precise.
As other people have commented, there are other specific kanji for 「みる」. The two common ones (also listed in the dictionary entry above) are:
- 「診る」, which means to examine, in the sense of a doctor examining a patient, and
- 「看る」, which means to watch over, in the sense of a nurse watching over a patient.
If you write
people might think you simply meant "the doctor looked at the patient" in the sense of "glanced casually at the patient". On the other hand, if you write
it's clear that you mean "the doctor examined the patient".
There are even rarer kanji. I found a blog which mentioned 「覧る・視る・覗る・監る」
as other possibilities. These are rather obscure, and if you make a habit of using them, your Japanese sentences will start to look strange. On the other hand, learning that all these kanji have a meaning related to seeing could be useful for studying kanji compound words. If you want to find out more, a kanji dictionary is a good place to look.