The GDPR does not apply to records kept for personal purposes. If you want to scour the web on a regular basis and write down all the names of criminals in a notebook, then you are still free to do so.
Employers can already request that potential employees hand in a certificate from the authorities that they have no relevant convictions in Germany. (Führungszeugnis) All convictions past a statutory limit are not listed. There’s an extended version for especially sensitive positions (working with children, security sensitive positions,...) that lists convictions that are no longer part of the normal official limit.