I would clean everything you can get at. Then regrease where neccessary
Take the blocks out too, clean and grease up the threads on the long allen bolts.
I would take the pads out before applying the brake cleaner.
Guess your bike is a curvy, so you have two pistons in the rear brake. 1 each side. Don't push them in until you have cleaned all the crud off.
If you are going to get new pads using a screwdriver or similar on the old pads to push the pistons in is favourite.
Have fun. And get used to it. You'll be doing it often.
Doh! It's at least 8 years old. Of course it's a curvy.