Eric posted while I was responding as well, I would do what he says.
Here is my original reply:
Before deciding to brown the lock, have a look through old posts on the subject, and pictures of originals. It also depends on what style of rifle you are building.
I won't claim to be anywhere near an expert, but I have been taught that the locks were case-hardened. Remember, they were made from iron, and needed some sort of hardening - hence the case hardening.
I think that the appearance of browning on antique locks is likely just rust over time instead of deliberate browning. Again, depends on what style of gun, and what time period.
You can certainly brown the lock if you want to. No, you don't brown the face of the frizzen. You could heat-blue, case harden, or charcoal blue (not intending to start another discussion of the differences) the cock jaw screw. Lots of folks do some variation of that.
Cheers,
Norm