You can buy a whole beef tenderloin for a small fraction of the per-pound cost of prepared tenderloin cuts like filet mignon. The trade-off for the cook is more labor.
Most of the whole tenderloins you can buy are vacuum-packed PSMOs (pronounced "pismos"), which stands for "Peel and Side Meat On." A PSMO tenderloin is three cuts of meat held together by connective tissue and covered by a layer of fat. Peeling is the process of removing the fat and connective tissue, called silver skin for its sheen.
Place the vacuum-packed package in the sink. Slit the plastic with the knife and remove the meat. Rinse the meat in running water to get rid of excess juices and pat it dry with a kitchen towel or paper towels. Move the meat to your cutting board.
Slide your fingers under the exterior layer of fat and pull off as much as you can. A weak membrane holds the fat to the meat; you should be able to get your fingers underneath that to easily free the fat. You might need the knife in a spot or two.
Separate the the primary tenderloin muscle and the side meat, also called the chain meat. Open the seam between them with your fingers and cut through it with your knife. If you're good, the chain meat will come off in one long piece.
If there's fat left on the tenderloin where the chain meat attached, remove it as you did the exterior fat.
Save the chain meat for another recipe. It works well in steak sandwiches.
Remove the silver skin, which is tough elastic tissue.
Scrape off any muscle that covers the silver skin. Save that meat with your chain meat.
Carefully slide your knife under the silver skin so it's perpendicular to the length of the tenderloin. Do this about a third of the way from the narrow end of the meat, with the sharp edge of the knife facing the thicker end of the meat.
Angle your blade slightly upward against the silver skin so that as you slide your knife, you leave as little meat as possible with the silver skin. Slide your knife until it pops out the thick end of the meat.
With a towel, grab the loose end of the strip of silver skin you just released and pull it taut. The towel helps you get a good grip. Slide the knife under this strip toward the narrow end of the meat to free the full strip.
Repeat until you've removed as much of the silver skin as possible, and you're done.