Deep snow and frozen ground make digging a cathole impractical. Instead, do your duty near the snow’s surface, where the freeze/thaw cycle will help break down the waste.
Hike at least 200 feet from camp, to minimize your impact, and choose a south-facing location (sun speeds decomposition).
Substitute snowballs for TP (which must be packed out). Dry snow? Bring some packed-down snowballs from around camp.
Use separate pairs of gloves for cooking and crapping to avoid cross-contamination.