For most people a good quality down jacket or vest will suffice. For the naturally chilly or for very cold winter trips, you might consider down-insulated mittens and pants as well.
The most important skill when you step up to multiday trips is route finding. A stocked hut won't do you any good if you fail to find the thing. Experience and training are the most important components of overland navigation, an art well beyond the scope of this article. Look into a course and read a book on map and compass.
Backpacking With Skis
You reach out to catch yourself, your arms posthole in up to the shoulder, and the pack drives your face two feet into the snow. Such are the pleasures awaiting the novice ski backpacker.
But, if you can just get over that whole face-smashed-into-the-snow thing, there's nothing more rewarding than being fully self-supported, cruising through the truly empty mountains.
Skills. As mentioned above, skiing with a backpack is unbelievably difficult at first. You'll need to fine tune your style, learning to keep your weight more upright and turn less aggressively. Those bounding turns of your unburdened, lift-served day trips will wear you out in a matter of minutes under a full pack. Half the battle is learning to fall sideways instead of face first.
Cooking outdoors without the benefit of a stable platform and seat requires some getting used to. Even if you're only camping in a spot for one night, digging out a kitchen is usually worth the hassle. Find a hill and dig into the side, building a counter and bench. You can throw your sleeping pad down to sit on. You should also bring a one-foot square patch of old, closed-cell foam for your stove--without it, your stove will melt its way through the snow (a shovel blade can accomplish this in a pinch).
Additional gear. The equipment required to safely camp in subfreezing conditions and deep snow is not insubstantial.
First, the backpack. The importance of having a well-fitted, internal-frame backpack for skiing cannot, repeat cannot be overstated. When skiing, you want your load to feel like a part of your back; only the better internal-frame packs can genuinely accomplish this.
Second, shelter. There are two schools of thought: some believe snow caves are better and others stand by tents. While it is true that a snow cave will keep you surprisingly warm, they are nothing less than a colossal pain in the neck to build. For a beginner, it can take hours. Even for an expert it is never anywhere near as quick as putting together a tent. The amount of work you'll put into building a snow cave or igloo virtually never outweighs the hassle of bringing a tent along with you, in my estimation.
Having said that I do recommend building a snow cave a few times. It's a fun novelty sleeping in one and if you ever get stuck out in the open, you should know how to build one with reasonable efficiency.
Third, cooking. You'll want a serious stove, one that melts water fast. I recommend sticking with a white gas stove for their blast furnace output. You'll likely have to melt all of your drinking water, so make sure you have a large capacity pot, about twice as much fuel as you would normally take and lots of patience. A garbage bag dedicated to gathering clean snow for melting is also useful - the snow near camp tends to get kind of dirty.
Morale. With trips like this, assuming there's no guide, it's best to go with experienced friends. Old hands provide an immeasurably valuable margin of safety and morale. Particularly after the sun goes down, the cold--even when felt by well-clothed, dry people--can have a surprisingly drastic affect on good cheer. Once you stop skiing and encountering new terrain, it takes a certain amount of experience to keep up your spirits.
Staying active puttering around camp helps tremendously, particularly if that puttering results in hot drinks. So does conversation and a deck of cards. Needless to say, a bottle of Southern Comfort also works magic on your group's level of feistiness, but don't overdo it. There are few things as unpleasant as watching the contents of your stomach melt the snow between your feet, believe me.