One of the many ways you can build backlinks is by commenting on other blogs. There's nothing quite like it when you get the conversation going. To start off creating a .gov and .edu backlink profile or your blog, start finding such blogs to comment on. A great website that'll help you do that is Drop My Link. You can search for blogs based on the desired keyword. But more importantly, you can select the type of blogs you want to see from the drop-down list. From there, you can choose .edu or .gov blogs, and get the relevant blogs based on your keyword!
You can then filter out your results. Go for blogs with a high PR first, because commenting takes time, and you don't want to waste time on low ranking blogs.
Aside from comments, getting a proper dofollow backlink from .edu and .gov blogs can be very difficult. One strategy is to feature some such bloggers in a post of yours, so that they'll link back to you in return out of gratification. For example, you could create a list of 'the best university bloggers', or 'the best students blogs in the U.S' etc. Small recognition awards like these are sure to get attention, and hopefully a backlink.
To find such blogs, here's a Google Search trick. If you search Google for <site:.edu> (without the angled brackets), you will run a search for all domains with a .edu extension. You can also search a specific site using this method <site:yoursite.com>. If you search Google for <inurl:blog> (again, without the brackets), you will land all webpages with the term "blog" in their url. You can combine the two to run a search for .edu blogs - <site:.edu inurl:blog>. Moreover, you can add individual search terms within quotes to refine your results according to keywords <site:.edu inurl:blog "technology news">. This way, you can feature 'top university technology blogs'
If your website provides useful information on a particular subject, then you may use that to get inside a university's resource page. Resource pages are there for linking out to other sites where students can find useful information regarding a particular subject. So if you are a computer guru and know your way around Linux development for example, you could contact computing institutes' web team to put out a backlink to you.
You can use the same search trick you used earlier to find resource pages. For example, run a search for <site:.edu inurl:resources "technology"> or <site:.edu inurl:links "Linux"> (without brackets), and you'll find out plenty of universities you could contact.