How do I properly support the header when expanding a door opening?
From what I can see here, and assuming not a load-bearing situation, I'd do this:


Remove the existing header. You may need to use a recipro saw to cut fasteners behind the plaster on the room side.
Since you have access from the back side, run a new, full-width double 2x6 header. If you don't have trimmer studs or other support available, fasten the header into the king studs with lag bolts. I'd probably use 3/8" x 4", properly pre-drilled and countersunk. 
Toenail into the pin studs to anchor them to the header. 


Optionally use construction adhesive where needed for additional support.