Outlet Relocation and wiring
Your plan sounds good, except for one thing you do have to check.  

Anytime you remove an outlet, make sure you are not falling below the minimum outlets required.   In particular, a 6-foot cord from an appliance anywhere along the wall needs to be able to reach a power outlet.  It appears your old outlet is near a threshold, so if it's moving farther from the threshold, that is fine.