How can I use a servlet filter to change an incoming servlet request url from
http://nm-java.appspot.com/Check_License/Dir_My_App/Dir_ABC/My_Obj_123
You could use the ready to use Url Rewrite Filter with a rule like this one:
^/Check_License/Dir_My_App/Dir_ABC/My_Obj_([0-9]+)$ /Check_License?Contact_Id=My_Obj_$1
Check the Examples for more... examples.