Fully disassemble the gun, clean it up so you know you are looking at the metal. Study it carefully and plan out your work. When you are done, you want the gun to look the same as when it left the factory.
Make your own sanding block about 1-1/2" wide by 3" long. This is small enough that it will lay flat on the flat surfaces and not tip. Do not use power sanders, files, or any other tools except the flat sanding block.
If you look at the factory finish, it will likely have a grain direction from front to back. Always sand in one direction for all the parts, do not cross the grain once you have laid it down. Curved surfaces are polished with the sanding block also, then carefully blended with a strip of cutting cloth.
I always start with 150 grit backed by a solid sanding block. Once you can no longer see pits or scratches at 150 grit, drop to 180, then 220, then 300. At this point, it is ready for a light buff and bluing. If you are going to have the markings re-cut by an engraver, this is the time to do it.
For a really fine job, you can drop all the way to 600 grit and wet sand it with oil, but that is an awful lot of work. Metal finished like that has a glow to it you cannot get any other way. If you want to know what it looks like, look at a 1970's Browning Hi-Power pistol.
If this sounds like a lot of work, it is. If you would rather pay someone else to do it, find a really top-notch gunsmith and let him refinish it. Because of all the flat surfaces and straight lines, the Colt Hammerless Pocket Autos are really nice when redone properly, and really sad when poorly done.
You might be able to find a replica of that grip, or even an original.