I don't have anything but a passing knowledge in this area, but what you are look at is called search engine optimization (SEO). There are companies that specialize in SEO. I am not up on their current methods, but I suspect they change methodology frequently as Google and the like will want to prevent SEO companies from manipulating results. SEO essentially allows companies to boost their page ranking without paying for ads to appear at the top of search results. Do some searches on SEO or contact a company that does SEO and see if they will talk to you about their techinques.
One technique that probably still works is to create pages on another domain with the search terms you want with links to the target page. So, create reviews on a 3rd rate product review site that contains the name of your client and in that review post a link to the competitor's site. You may be able to check for this by simply searching for pages that contain your client's name and a link to the competitor's site.
Whether any of this is illegal, anti-competitive, fraudulent, or ethical is an ongoing question. There are valid arguments on both sides, but I'm of the opinion that SEO is at the very least unethical, depending on the methods employed.
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