Check Existing Back Links
First, I check the client's existing back links. This helps me get a sense of whether or not there might be some "link issues" (i.e. excessive spammy links that could be limiting site performance).
It also helps reveal opportunities for future link relationships (i.e. these people, organizations, businesses, etc have already linked to client's site, they'll likely be willing to link again).
Checking existing links also identifies sites that might have linked at some point in time, but for whatever reason, aren't linking any longer.
Check Existing External Links
In addition to checking for existing links that point to a client's site, I also check to see where my client has already linked out to. Again, this helps show any existing relationships that might be useful in future SEO marketing campaign efforts.
In some cases, due to bad advice, we will find that our client's site is doing excessive external linking. This usually happens because someone told them that it was a good idea to "link out" to a bunch of relevant sites (hint: generally not that great an idea).
Finally, checking existing external links can identify site parasites, hacks and other nastiness (i.e. someone hacked the site and is now linking out to their spam pages).
Check Competitor Links
Once I have a sense of who the client's major online competitors are (note this can be different from perceived offline competitors), I will usually run them through a variety of back link checking tools.
Most of the time (this is especially true for law firm websites / blogs) there's not much to be gained here. This is due to the fact that most law firms are participating in crappy SEO techniques.
Nonetheless, checking competitor sites that perform well in organic search results for your target terms provides a good "lay of the land" for forecasting how difficult certain results will be to achieve.
Check External Links for Relevant Authorities
Finally, and probably most useful, I use Screaming Frog to crawl relevant authoritative sites for external links. This helps identify sites that are likely to be good candidates for obtaining high-quality links in the future.
This is also a great way to find low-hanging link fruit (i.e. broken links, link pages on high-quality sites, etc).
It also helps to identify sites that don't link out as a general policy. Sites that have a policy of not linking go on the long-term link list.
Admittedly, none of these techniques is really cutting-edge. Most experienced SEOs have been using these 3 basic techniques for years now. The reason that I bring them to you attention is that most of the law firm websites I review, haven't even performed these basic checks.