Koreans did not have a writing system before Chinese characters were adopted.

The earliest writing in Korea of a non-Chinese language comes from an archive of mid-sixth-century provincial administrative documents, first deciphered in 2017 (see 이승재, 木簡에 기록된 古代 韓國語, pp. 180—191). But it represents Korean grammatical morphemes with similarly pronounced Chinese characters.^1 For example, in the archives we have the phrase:

不行遣乙白

In Chinese this means

not go send bird report

But we know from context that the phrase must mean something like "report that they did not go." Then we can look at Late Old Korean from the eleventh and twelfth centuries, and understand that 遣 (ancient Chinese pronunciation *kʰen) must have written the ancestor of the LOK subject honorific suffix *-kye and 乙 (ancient Chinese pronunciation *qrət) must have written the ancestor of the LOK accusative case marker *-r.

So we know for sure that this language is an ancestor of LOK and thus of modern Korean, and that

不行遣乙白

is actually to be understood as

not go**-HON-ACC** report

And thus

"report that [the honored ones] did not go."

All Korean sources from before that yet discovered are written in Classical Chinese, with few clear vernacular influences. There was no indigenous Korean writing before Chinese.

For a general overview of techniques used by the earliest Korean scribes to write Korean, see Zev Handel's Sinography: The Borrowing and Adaptation of the Chinese Script (Handel 2018, Brill).

^1 Or with Chinese characters whose translations into Korean sound similar to Korean grammatical morphemes. The English equivalent might be writing the phrase "to be" with Chinese characters for the number "two" and the insect "bee."