You can create this look either at capture or in post.

In-camera: shoot when the weather and light already support it—dark clouds with low sun/sunset can naturally give a cool dramatic sky and warm skin. To push it further, set white balance so the background goes cooler than normal, then light the subject separately with flash fitted with a warming gel. The ambient light exposes the sky; the gelled flash warms the subject. Balance flash and ambient exposure to control how strong each effect appears.

In post: edit the sky and skin separately. In Lightroom, first adjust the overall image or sky to get the cool grey/green tone you want, often via white balance and color adjustments. Then use an adjustment brush or local mask on the subject/skin and warm it back up to taste. In Photoshop, use separate layers for the different looks and combine them with a layer mask so the sky and skin can be adjusted independently.

The key idea is separate control: cool the background, warm the subject.