Nowadays a Chinese person seeing a character for the first time can look it up in a dictionary (it doesn't matter how) and get the phonetic or Pinyin transliteration, so that the pronunciation is obvious.
How did people get the correct pronunciation of unknown Chinese characters without Latin or phonetic alphabets? I mean, I would expect the language to be "self-contained", so to say, without resorting to external alphabets.
The pronunciation of characters was glossed using the Fanqie (反切) system, which uses two existing characters whose pronunciations are known to determine the pronunciation of the unknown character.
Suppose that I wanted to know the pronunciation of「東」. Looking this character up, I'd see that it was phonologically glossed in dictionaries as 德紅切, which means to take the onset (basically the initial consonant) of the first character, in this case「德」, and the "final" (which really is everything after the onset) of the second character, in this case「紅」. In Pinyin,「東」is dong, and reconstructing this syllable with the fanqie system we'd get the initial of「德」(Pinyin de) and everything after the onset of「紅」(Pinyin hong) to get dong.
Note, fanqie dictionaries were compiled by scholars in consideration of all the varieties of Chinese, not favouring any particular topolect, which means that they work equally well for all these varieties. In addition, they also work for Sinoxenic pronunciations of Chinese-originated vocabulary in Japanese, Korean, and Vietnamese. This is possible because the two characters used to construct the pronunciation are not phonetic symbols like alphabet letters, but have their own unique pronunciation in all the varieties of Chinese (as well as Japanese, Korean, and Vietnamese, due to centuries bordering on millenia of cultural and language diffusion across East Asia). The caveat is that the phonology guide is a good approximation but not always an exact match, as the sound evolution of the different varieties of Chinese from the ancestral language is not always consistent everywhere.
This requires the reader to be somewhat literate already, of course - but then again, which dictionary in any language doesn't?
Demonstration that it is a good approximation for other languages which have used Chinese characters:
東, 德紅切:
In ancient China, there're two ways for phonetic annotation mainly.
读若法
用一个汉字来注另一个汉字的读音方法。(Use a character to annotate another character's pronunciation.)
比如：儡，相败也，……读若雷。（《说文解字》）(e.g. 儡 is pronunciated as 雷.)
反切法
用一个汉字或注音符号表示“声”，用另一个汉字或注音符号表示“韵”和“调”，把它们拼合成被注字的读音的方法。(Combine one character's consonant and another character's vowel to annotate the pronunciation.)
比如：缓，胡管切。（《广韵》）
“缓”字的读音，就是取“胡”字的声（h）,取“管”的韵和调(uan),然后拼合成（huan=h+uan）。(The pronunciation of 缓 is combined by the consonant of 胡 (i.e. h) and the vowel of 管 (i.e. uan), then huan=h+uan.)
See 古代注音方法
I don't know specifically about Chinese, but one thing all researchers into historical phonology can use to help them is rhyme. Where two words in a poem or song no longer rhyme where clearly they were meant to when authored, it is an indication of sound change, either in the past, or between dialects.
