There are a few different methods that are used to discern how words used to be pronounced. For example, let's look at Shakespearean English vs. modern English.

The first and most reliable method would be looking at what people of the time wrote about their language. These primary commentaries tell us how certain words sound, which words rhyme, etc. Ben Jonson, an english dramatist who died in 1637, wrote a book titled "English Grammar" where he discussed some of the intricacies of english pronunciation. For example, at the time if there was an "R" preceded by a vowel, it would be sounded like a drawn out, almost pirate "arrr." He describes it as a "doggie" sound, or a sort of growl.

Spellings are the second type of evidence. At the time of Shakespeare, spellings were a much better guide to pronunciation than they are today. Words were pronounced as they were spelled back then, and because there wasn't too much in the way of standardized spellings, it can tell us a lot about how words would have been pronounced. Of course in modern times, we have new pronunciations but we kept the old spelling. This is in contrast to languages like German, which have periodic spelling reforms to keep up with changing pronunciation. This is part of the reason why words like knead, bread, wear and great all have different vowel pronunciations, despite sharing vowel spellings.

Another way is to look at texts that are intended to rhyme, but often don't in modern english. For example, if we look at the last two lines of Shakespeare's 116 sonnet,

"If this be error and upon me proved,

I never writ, nor no man ever loved."

Now in sonnets, these last two lines are supposed to rhyme, but they don't in modern english. Thus we can infer that "proved" was pronounced so that it would rhyme with "loved" or vise versa. The trick then is figuring out which pronunciation changed to match the other. That's where the other methods come in. It turns into a bit of a puzzle, but it's doable. I believe that there are production companies who perform 17th century plays in their original pronunciation.

What kinds of time depths are you interested in? If the state of the language is significantly old enough to have descendant language varieties you could use the comparative method among the proposed related languages to determine what features (including not just phonological features, but perhaps even some prosodic features) the ancestor language variety might have had. If there are written samples going back some time, then so much the better! If there is writing in the language, then perhaps ques such as rhyming can help you narrow down the phonological forms of the older lexical forms. This technique has been used for Middle and Early Modern English (with the aid of poetry like Shakespeare) and Middle Chinese alike (with the aid of old rhyming dictionaries). Another source might be meta commentary on the language itself, for example a journal entry noting perceived oddities in the speech of others, ex. "they say [part X] of [lexeme Y] like [lexeme Z]". That kind of commentary gives a basis for comparison for your modal of the past phonological forms of a lexeme (or lexemes if you are proposing regular sound shifts).

Those are best known tools at the historical linguists' disposal.

Op, do you have a specific language in mind? If so, there have been several posts on pronunciation of, for example, Ancient Egyptian, or Latin, or others that may be of interest.