Add a style (CSS) to a custom block
Easiest way to add a custom class to a block would be to override the default block.twig.html twig template within your theme.

You'll need to copy the above file from /core/themes/stable/templates/block into your custom theme using the same directory structure.

**  NB don't edit the file found in core directly!! This is bad practice and you'll lose any changes when you upgrade Drupal core. **

Clear the cache and you should now be able to edit the block template file and customize it.

Then, to load CSS, you must define an asset library. You can then load that asset library using {{ attach_library('mytheme/special-block') }} in your Twig template (assuming the theme is called mytheme and the asset library special-block).

Helpful links:


Working with Twig templates
Template naming conventions
Debugging Twig templates
Adding stylesheets (CSS) and JavaScript (JS) to a Drupal 8 theme