Troubleshooting a leaking exterior door
Verify that the door is leaking and not some other place. Have a helper with you and on a dry week spray with a water hose against the door where it meets the jam see if any water leaks in from that. Give it 30 min to show up and if that doesn't work try spraying around the outside of the frame and see if it leaks in then. 
Follow the stains to the wall. It is entirely likely that the leak is somewhere else and the door is the first break in the seal so it leaks there. You may need to pull moulding and or wall sections around the door. Following rusty nails can do it to.  I have an addition to a mobile they did not keep a solid pitch so the add-on is about 4" below the main roof.  If ice starts damning it will go down under that roofing all the way across the roof and in to the outside wall 16' away before leaking out. We had a bay window dripping last winter because of that.