I want to dye this cloth vermillion.
The kanji and the word are exactly the same. That means they share meanings as well. You can also say it's scarlet red or blood as well, though more in a poetic sense, I'd say. I don't think you'll hear surgeons calling blood 朱.
The reading あけ is like the beginning of 開ける, which means "to open." Just imagine you're opening (あけ) a person up, and out gushes out a vermillion colored blood. You shouldn't open people up like that.