It is widely accepted that .edu and .gov backlinks carry a larger weight for search engine algorithms. Despite the fact that Google has never admitted it (sometimes it even neglected) most SEO and online marketing experts would confirm that .edu and .gov links outperform other extensions in terms of search ranking juice. There are several ways to collect those links, below I will explain two of them, depending on whether your blog has relevant content to educational and governmental websites or not.
Blogs with relevant content
The first step to get some .edu and .gov backlinks pointing to your website is to locate sites related to your topic ending with those extensions. Using the Google operator site:.edu and site:.gov combined with the keywords of your blog should enable you to find an extensive list of such sites.
site:.edu "keyword 1"
site:.gov "keyword 1"
Once you have a list of sites explore them and select the ones you think would have a higher probability of sharing some external links with your blog. Finally contact the webmaster of those sites and point them to the content on your blog that could be relevant and valuable to those sites. It is always a good idea to link first to the sites before asking the backlink.
Blogs without relevant content
If your blog does not have content that could be relevant or valuable to educational and governmental websites your should look for types of sites that allow users to create a backlink without interacting with the webmaster, and the most common types of those sites are blogs and online forums.
Again you can use the Google operators to generate a list of such sites. Notice that the site:.edu argument limits the search results to sites with a .edu extension while the inurl:blog attribute ensures that results will have the word blog on the URL structure.
site:.edu inurl:blog "keyword 1"
site:.edu inurl:forum "keyword 1"
site:.gov inurl:blog "keyword 1"
site:.gov inurl:forum "keyword 1"
Finally track down the sites with discussions relevant to the topic of your blog and join the conversation, leaving a backlink to your site either through the comment section or on inside a forum thread. Bear in mind that you should not spam those sites for the sake of getting as many backlinks as possible. Make sure that you are adding useful information and not just polluting the online environment.
Update: Check out the “Enhanced” version of those search queries.