It's good question to ask "How to check the broken links" ! There are various way to check out broken links links as well as sources too but I do prefer online tool to check out broken links ! I have been using deadlinkchecker.com for long to & it's best online tool to check broken links !
There are many tools available to check entire websites for broken back-links. Screaming Frog is the best tool for broken links.
Please refer these FREE tools :- brokenlinkcheck, deadlinkchecker, Google webmaster tool, W3C link checker etc.
Webmaster Tools is your best bet. If the site isn't YOUR site, though, and you can't access GWT info, then plug the site into Screaming Frog. Then sort your links (internal, inbound and outbound) and start checking them manually. That's a great way to find opportunities to request backlinks by showing site owners that they have bad links on their sites.
Screaming Frog is the best! And like you said, broken links are just one way it can help you improve your site's SEO
Since everyone here seems to be on board with Screaming Frog, I thought this might be worth sharing: http://www.seerinteractive.com/blog/screaming-frog-guide/ This guide by Seer Interactive (read: NOT MY COMPANY, NO AFFILIATION) is really helpful, especially for anyone who hasn't used the tool before.
Here is how to find broken links using A1 Website Analyzer: http://www.microsystools.com/products/website-analyzer/help/site-analysis-seo-audit/#linkdata You can probably find similar guide for whatever website crawler / website audit / link checker tool you are using
All the above is good for finding broken links on a site, but if you want to find the broken links on an individual page (perhaps for link building purposes) then use a Chrome or Firefox 404 checker add on - it scans a list of links on a page, then highlights the broken ones...very useful! All the best
If you want to check all of your broken links then you have to get subscription on Ahrefs.com to fix all those things. You can also try majesticseo.com