You are looking for what is known in the industry as reputation management. There are many businesses with experience in this area that will work on this problem for your client.
If your client is in Europe, it may be possible to use right to be forgotten laws to force search engines to remove pages from those search results.
In the rest of the world, it is not usually possible to control the placement of a site that you don't own within the Google search results. Most of the time, the best way to deal with an unflattering search result is to provide better, more relevant results that drown it out and push it off the first page.
Social media sites are often used as part of that strategy. Facebook and LinkedIn often rank very well when you create public profiles with lots of (more flattering) details. Just make sure that the two don't contain the same information which would cause Google to show only one of them.
A personal website is another good way to get a result higher up in the search results.
Profile pages on participation sites (such as StackExchange) are another good way to get great content to the top of the search results. If your client already participates in some of the sites, get them to use their real name on their profile. Some industries have many such forums and sites that you could suggest your client partipate in.