Provided you are not breaking into any aircraft systems or structure, or making permanent changes, there is no requirement for any STC or issue of a CRS for the work to fit the go-pro or its mount.
I regularly use one both inside the cockpit and externally, I would reccomend a bolt down type clamp for external mounts, similar to the type of clamp used to mount ipad holders to control yokes in the fixed wing world - RAM mounts do some good products.