Do I have to state "[Emphasis mine]" every time I highlight the relevant part of a quote with formatting? Can I omit it at all?
It is not required.
Saying [emphasis mine] is simply a way to indicate that something you quoted is not entirely true to the original … or you might add it as a way to draw the readers' attention back to the relevant portions you highlighted for a specific purpose (i.e. parts that are not highlighted in the original quote).
As Robert Cartaino said, you don't have to include that note.
Personally, I usually include the "emphasis mine" in the preceding text. Watch as I quote what you said, emphasis mine:
Do I have to state
[Emphasis mine]every time I highlight the relevant part of a quote with formatting? Can I omit it at all?
Another option, depending on the structure of that preceding sentence, is to parenthesize the "emphasis mine" rather than setting it off with a comma.
Once you have one such note in your post, I think most people will infer that additional formatting later in your post is also added by you.