There is no way to do it without config cheats in insecure mode (or actual cheat clients), you can't do this while playing. There's a reason you only see this in frag movies. You could maybe get it working via preloading as it's just a texture replacement really but there's no telling how often it would break on Valve servers, on community servers it would probably work.
If you're working on a frag movie, the easiest way is to just use Lawena (a glorified config loader) which has a menu for changing skyboxes (this does not work in normal games!). And if you use HLAE, do not turn off insecure mode and try to connect to normal servers, there's a risk of getting banned.
Also some community maps internally contain the skybox instead of referencing a skybox within TF2's base files. This means that on maps like reckoner, normal skybox replacements do not work and you'd have to play on an alternative version of reckoner with no skybox. So even if you were to do this, it wouldn't work on all maps.