I don't know how to change the colour of the sky and it would seem slightly odd if the sky were grey at all altitudes even above clouds.
However it's possible to create overcast weather by having mostly white CloudsNoise.tga (full white is perfectly even cloud, with some detail it looks better) and full white Clouds256.tga (these are the default textures; you can make custom texture names and edit them into map's config file).
It isn't perfect by any means, and the game will still think that the sky ought to be blue, and you end up with some strange things. It will not affect lighting, it will mostly just look like thin cirrostratus cloud and the Sun will shine through it just as brightly as ever. Regardless, it can create a passable impression of overcast sky.
Here's a basic impression on how it looks on Rain/Snow weather on Crimea, cloud level 3000m, 17:30 clock. The three first images are done with completely white CloudsNoise.tga, creating a spooky, fully featureless grey sky.
Below clouds:
Climbing into the cloud coverage:
Above primary cloud layer. You'll notice the "edges" of the clouds are blue-ish, which would blend perfectly with the normal blue sky, but not so much with overcast high clouds. Not much that can be done about it... although perhaps changing the sky colour itself could possible fix this...
On this image, there's some diffuse noise on the CloudsNoise.tga file, but only slightly darker than white, and weather is set to Clear.