1) The first thing you need to do is capture a replay. It doesn't have to be anything spectacular, there are all kinds of surprise little poses, angles and moments to be found even in replays where you didn't kill anybody. If you feel like a life has screenshot potential, capture a replay of it.
2) When you exit the match and return to the main menu you'll want to open up Advanced Options, scroll all the way down and make sure Save Screenshots to Steam Cloud is unchecked. For whatever reason having this box checked won't let you take uploadable Steam screenshots while you're in Replay, and if you hit the screenshot button and it doesn't make the "vweet!" noise this box is probably checked. Nothing is more frustrating than finding the perfect moment, lining up the perfect shot, hitting the Screenshot button and not having anything happen! You will also need to open up your normal Options menu, under the Keyboard tab there is an Advanced button, click this and make sure Enable Developer's Console is checked. This will be important later.
2a) There are actually two settings for screenshot keybinds, one in your normal Options for TF2 Screenshots and a separate Steam Screenshots keybind, which can be found in your Steam menu -> Settings -> In-Game tab. It is safe to bind both of these keys to the same key so you know no matter what you're taking the "right" kind of screenshot. Screenshots with metadata that upload to the Steam cloud come from the Steam Screenshots setting, not the TF2 one.
3) If you run your game on low settings, it might be ideal to go into your Video options and boost the graphics for this, or run the client with a more recent Direct X setting than dx8 or 9- since you're not actually playing in a match and the replay is entirely rendered client-side things like low FPS won't make a difference if you're looking to just take a pretty snapshot. You can always reset your settings before playing another live match.
4) This is the fun step! Run your replay, play through it and look for an interesting moment to capture. Running the replay with the third person camera is a good way to get a sense of what is happening around the subject you're following. If you rewind to re-examine a scene hit the third person camera button again to reset and restore control of the camera, otherwise it'll be locked in the path you just sent it on. Things you could look for are interesting poses; ragdolls; near-miss rockets, pipes or Huntsman arrows; a fun jump or even just a neat kill. Having a good eye for "moments" will help you find an interesting subject.
4a) A notable error with the Replay system: sometimes when you load up your replay and click into third person your player model will be invisible. The fix for this is to let it play for a second in third person, pause, set an Entry point, play again, pause and then hit the Rewind to Entry Point button. This should force Replay to render your model, hats and everything. It's a bit of an old bug but it's worth pointing out here so you know how to handle it when it comes up.
5) Once you've found your moment and the camera is paused around it (spacebar by default) switch over to the Free Camera to find an angle and compose your shot. Two important camera settings- which pop up when you mouseover the Free Camera button in the bottom right- are Speed and FOV. You'll want to lower your Speed so you can inch the camera into position without a keystroke sending it sailing in that direction. Widening and lowering the FOV broadens or narrows your depth of field- it's generally better to lower the FOV because it creates much greater focus on your subject and makes the kind of screenshots people will think you took in Source Filmmaker. A low-FOV depth of field can be used to capture two distant subjects when they're lined up in the frame- like a Sniper's victim and the Sniper himself, or a Scout and his Sandman ball, or a Spy stalking his prey. Play around with these settings and find a shot you're happy with! Experimenting is half the fun and this is the step to do it in.
6) With your shot composed and your camera positioned, open up the developer console (~ by default) and type "cl_drawhud 0"- what this does is it disables your HUD, which means all the black vignette borders and camera lines will disappear, leaving you with a nice clean image. Once all your ducks are in order, hit your Screenshot key and snap a picture! If everything works out it should make an audible "vweet!" sound and a little window pops up in the corner of your screen saying the screenshot has been successfully taken! This is good, it means that screenshot is eligible to upload to the Steam cloud and share in your gallery for all your friends to see. With the HUD disabled feel free to move the camera around and try snapping shots from multiple angles. If you need to make camera adjustments "cl_drawhud 1" re-enables the HUD so you can click your camera buttons again. It's that easy!
7) The last step of the process is to close out your replay and bring up your Steam Screenshots window- it usually pops up on its own once you close the client but you don't need to close the whole game down to access it. Pick and choose which of your shots looks the best, give it a nice caption and upload it to the cloud! Boom, done. You did it! You've posted a beautiful screenshot to Steam!