In my previous trip 2 years ago, we walked the other way round, from Samcheon-dong towards Tosokchon. Either way is fine, depending on your schedule.
http://musingsofmissn.blogspot.com/2014/01/mr-missy-in-korea-day-13-28-may-2013.html
Samcheon-dong
We took a stroll from Tosokchon, back to the main gate of Gyeongbokgung, and along the palace walls on the Eastern side. Then cross the road into Samcheon-ro street, marked in the above map with a red line.
Alternatively, you could take subway for 1 station from Gyeongbokgung to Anguk and begin the exploration of the area from Anguk Station, Exit 3. That was our starting point of the Bukchon+Samcheondong stroll during Mister exchange student days.
Samcheon-dong is a quaint area dotted with many unique museums (Owl museum, anyone?), cozy cafes, organic beauty shops and pretty boutique stores.
Spring was admittedly more beautiful. This was taken 2 years ago in May 2013.