I upgraded SharePoint 2010 to 2013 and the Org Chart / Organization Browser no longer appears under MySite. Or clicking on a user (people search webpart successfully implemented) forwards to the SharePoint 2010 environment. How would I implement Organization Browser, have it appear under users MySites and forward to the SharePoint 2013 environment?
It's by design!
the Organization Browser is removed/hidden by design in SharePoint 2013.
To enable it back,
Try to browse this URL
Or enable the Portal Layout feature by runing the below cmdlet
Enable-SPFeature –Identity PortalLayouts –url http://<my site> –force
With SharePoint 2013 organization chart is not turned on by default.
After you enable the Portal Layout feature as Mohamed suggested， each user can browse to their MySites and then click on Sites from the top Right hand corner and choose to customize this page->Add Web Part -> Under Social Collaboration ->add Organization Browser web part.