I would recommend not focusing on improving your domain authority (it is ok to use for comparison sake) and focusing in more on organic traffic and conversions. There really is no answer that X number of backlinks will lead to X traffic increase in X amount of time for every client. There are hundreds of outside factors that can influence how quickly you will see positive results.
I have seen backlinks help push up rankings in as little as 1 month and some cases as long as 6 months. No two backlinks are created equal so the quality, niche relevancy, the competition in your niche, and your content will all be variables in this. Competition especially is a big factor. I recommend doing a full-scale competitor analysis to help determine the quantity and quality of backlinks you will need to rank in your specific scenario.
Here is my post to help that process: https://moz.com/blog/competitor-analysis-for-seo