Write a custom DAM handler that is called when an asset is uploaded:
https://helpx.adobe.com/experience-manager/using/damhandler.html
You need to read the user id using the User management API:
http://jackrabbit.apache.org/api/2.0/org/apache/jackrabbit/api/security/user/UserManager.html
You can get current user:
Session session = resourceResolver.adaptTo(Session.class);
UserManager userManager = resourceResolver.adaptTo(UserManager.class);
/* to get the current user */
Authorizable auth = userManager.getAuthorizable(session.getUserID());
Get the id and set the asset using AssetManger API.