Talking to the original author or publisher, searching news articles about the book in question, contacting Nielson BookScan about books published after 2000, and checking issues of Publishers' Weekly if the book in question is a best-seller are all possible ways to get the sales figures you need, according to the University of Minnesota. The publishing company is under no obligation to release these figures, however. This sometimes makes getting accurate numbers more difficult.
A publishing company or author may be willing to give you sales figures, depending on why you are asking. Likewise, a book that's sold well or has received a lot of publicity is more likely to have its sales figures reported by a news outlet or publishing magazine. Nielson BookScan is a subscription service that tracks book sales in similar fashion as Nielson tracks TV ratings. You are unlikely to get sales numbers from the service without a subscription, but it's worthwhile to contact the company and ask. However, it started in 2001 and does not have sales figures for any books released prior to that.
Using multiple sources is ideal, because sales numbers are often quantified over a certain period of time. As a result, you may see different sales figures for the same book from different sources. In those cases, compare the sales region the numbers are referring to and the date the information was provided.