I would like to add a new web part to my SP home site, showing the most active users on the site. Is it possible? And if so how?
There is no OOTB webpart available and I don't think this kind of data is available within SharePoint which you can consume. Still I would suggest you to check SharePoint Audit log reports, popularity trends and see if you can get report from this data.
Read this link for different options available for sharepoint usage reports.