It depends on the system. However, DisplayCAL should offer to install the profile after it’s been generated. You may still have to assign it to a monitor, which your DE should allow you to do. If you use X11 you then select either system profile or the specific profile in the various programs.
Here are some test profiles you can use to make sure everything works as expected: ColorWiki - Stunt Profiles
Elle Stone has some articles on the subject: Articles and tutorials on Color management