Should the website contain information which is discouraged to have in the CV?
In moderation, I don't see anything wrong with including content on your website that you would not include on a CV.  In principle it could hurt your career if you post something ridiculous or objectionable, but information about chess, ordinary volunteer activities, children, etc. doesn't seem problematic.  The main issue is making sure it doesn't overshadow the academic content: if you provide more details about your chess activities than about research, then it may give the impression that you are devoting more time and attention to chess than to research.  (Even if that's not the case.  For example, a grad student who is just starting with research but has a lengthy chess history might give the wrong impression, regardless of current priorities.)  If you want to cover non-academic activities in substantial detail, then it may be safer to set up a personal website that is clearly separated from your academic site.