It's dialectal. This started with the Chinese community in Southern Vietnam, chiefly in and around Saigon. Their home languages are Southern Chinese languages like Cantonese, Teochew and Hokkien, where 小姐 (Sino-Vietnamese: tiểu thư or tiểu thơ) is used as a common term of address for women, like madame or miss. In these languages, the 姐 character is pronounced with /ts/ - either /tsi/ or /tse/. As they brought this word into Vietnamese, it got written down as “chế”. Yes, they went with the pronunciation in their home languages, rather than the Sino-Viet one. Vietnamese doesn't allow the /ts/ consonant, so “ch” is the closest thing. It then spread to the non-Chinese people living next to them.
Note that it's originally used for addressing all women judged to be in the same generation as you, even strangers on the street. For your older sisters at home, Cantonese for example requires specification: 家姐 (gaa1 ze1 / gia thư).
Without knowledge of these Chinese languages, ethnic Kinh people mistakenly think that “chế” can only be used for one's older female sibling.
Proper or not depends on your viewpoint, but to me every dialect is proper.
The standard Vietnamese word for sister - chị - didn't come from this character, but most likely 姊 (SV: tỉ).