How you get a backlink depends on what kind of a website/webpage you are interested in. Once you gather backlink URLs for your competitors you will browse them and see what kind of sites the links are placed on, whether it's a forum post, a blog comment, a submitted article, an ad, or a paid blog post with a link in it. Judging by the way the link is obtained by your competitor, you can figure out a way to get the same (or a similar) link for yourself.
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Is there a way to keep track on all the requests that I assume you have to mail web masters etc?

You can do that using LinkAssistant by copying all backlinks from SEO SpyGlass workspace to LinkAssistant as linking partners. Just select them in SEO SpyGlass, right-click -> copy to clipboard, then go to LinkAssistant and click Edit -> Paste link partners. Then you will be able to send e-mails to the webmasters of those webpages and track backlinks from them.