Here's how the majority do it.
Play the game, record footage with film recorder.
When you want to start, you get a capture software like FRAPS (there are others, such as MSI Afterburner, DXtory, etc).
1) Start capture software (using FRAPS in this example).
2) Start film viewer, load film, get ready to play.
3) As you play the film back on your monitor, hit the HOT-KEY in FRAPS to start recording footage.
4) toggle on-off as needed to capture footage in several areas, also pausing as needed to set up camera angles in film viewer.
5) exit film viewer.
6) go to FRAPS folder where footage is saved.
7) review footage, decide how you want it to go together
8) use non-linear editing software (i.e. Sony Vegas for me), import/drop/edit the clips as you like -- don't forget sound and audio considerations
9) save project as you go, until you are ready to export.
10) export the segment you have created as a video file. Youtube has recommendations on their site as to what format to use. H.264 is their best suggestion if I recall, but I think WMV works well, as does DivX and XviD.
11) after render, check newly-rendered video to review and look for glitches/bugs you need to fix.
12) Upload video to YouTube.
Youtube will re-sample the program in their own format once you upload it -- that's why you have a delay before you can view -- but some formats re-compress better than others, so if you can stick to their suggested formats list.
Hope that gives you a very rough outline of the process.
