Most commented post should be the first post in the blog

by

Alex, the codex is your friend. You have a parameter orderby which you can set to order posts by comment_count. You can get the complete list of parameters accepted by orderby in the codex page about WP_Query

There is no need for any custom query, you can simply just alter the main query with the use of the action hook pre_get_posts. To order posts by comment count on the home and category pages, you can use the conditional tags is_home and is_category to target the homepage and category pages respectively.

add_action ( 'pre_get_posts', function ( $query ) 
{
    if (    !is_admin()
         && $query->is_main_query()
         && (    $query->is_home() 
              || $query->is_category()
            )
    ) {
        $query->set( 'orderby', 'comment_count' );
    }
});

EDIT

This code should be placed in your functions.php or any file that is related to functions.php, ie, any file used for rendering functionalities like functions.php does

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