you have to make 2 seamless textures, one for the upper-layer (opaque) and one for the lower layer (which has transparency)

and then you need to need to put both textures into one texture side-by-side
with the lower-layer on the left and the upper-layer to the right, like so


although of course your texture will have transparency.

save it as a PNG or TGA, and put it in ID1/textures.
then in-game you can load it by opening console and typing 'loadsky YourSkyName'

Or there is the old fashioned way:

Extract Quake.wad using something like Worldcraft.
Open Quake.wad in something like Wally.
Search around for textures starting with "sky_", copy and paste one naming it something else but keeping the "sky_" part.

Edit it. -mess around a bit with a couple, I have done blue skies with fluffy clouds or black with a few stars. I try to make them simple and not too busy but that's just my preference.

Make sure to remip & save, then save the wad file also.
Load up the wad file in the map editor your using and apply the sky like any other texture.
Also, within Wally you can load in the 8bit Quake.pal palette, that is,
if you want to use it with original Quake, otherwise do what talisa says.

Shambler234, you will find a tutorial and a lot of samples of higher resolution sky replacements in this forum. Please use the search function. Talisa even linked several in the Replacement-Compilation thread.

Talisa seems to confuse skyboxes with scrolling skies. There is no loadsky command for scrolling skies when talking about dp. Please do not mix static skyboxes with scrolling skies. The loadsky command only loads static skyboxes placed in id1\env subfolder (each skybox has 6 textures. Yes, it is a cube).

The scrolling sky texture replacements, which you want to use, are handled differently in supporting Quake engines:
dp uses the original (as it is used in original maps) texture format: upper and lower sky in 1 texture. Name them sky1 or sky4, as original id1 maps only have those 2.
fte and qrack use 2 different textures (for upper and lower sky). Name them sky1_alpha and sky1_solid and sky4_alpha and sky4_solid.

Notice that you can use different sky texture replacements for every map if you place them in map specific subfolder, like \textures\e1m1. They will override the ones in \textures for that map.

@henry
ah my bad, i thought the loadsky command also allowed you to load scrolling skies.
but i just tried it out and it doesnt work indeed for scrolling skies

disregard what i said then shambler234, and do as henry said