Block link with active class
It needs to be done with PHP. If you use the l() function to generate your links, they will automatically get the active class if you are the page to which the link is linked to.
To do this you would need to enable the PHP filter, but this filter is both recommended against, as well as non-existent in Drupal 8, so it would be better to create a block through code and use that instead of enabling the PHP filter.