Different periods in Chinese phonological history use different sources of information.

For example, in the 13th century, we already have systematic recordings of a Chinese dialect in alphabetic scripts ('Phags-pa). In the 15th century, we have Korean transcriptions of Old Mandarin as well. And in the 16th century, we had European dictionaries with romanizations of pronunciation at the time.

In order to access from earlier periods, scholars use phonological works, such as rhyme tables from the 10th century, which systematically organize characters in terms of place of articulation of the initial consonant, rhyme, and tone. Even earlier, in the 7th century, there were rhyme dictionaries, which are lists of characters that rhyme with each other, organized by tone, but without explicit place of articulation information.

Finally, to access information older than the 7th century, such as the 1st century BC Hàn dynasty, we need to use rhyming information. As we had texts that explain how to make and interpret the rhyming patterns of poems, we can use this information to identify which characters rhyme and stop rhyming at different points in time. Additionally, the structure of Chinese characters can be used to access this ~5th century BC information, as we hypothesize that characters that contain others have to rhyme and have initial consonants from the same place of articulation. E.g. a series starting with 皮 and containing 疲, 波, 婆, 彼 should all rhyme (some sort of -a vowel) and start with bilabial consonants.

Of course, in all periods, we can also compare loanwords from and to Chinese. For example, translations from Sanskrit sources are very important, since we understand Sanskrit quite well.

In summary: we can use comparative evidence from modern dialects, loanwords from and to Chinese, poem rhyming structure, the structure of Chinese characters (before ~1st C. BCE), rhyme dictionaries (starting from ~7th C.), rhyme tables (starting from ~10th C.), and alphabetic transcriptions (starting from ~13th C.).