Someone once accused me of being crazy for camping in Idaho when it was -25 degrees out. I called them a sissy. In other words some people get a kick out of camping in impossible conditions. This can be tons of fun if you know what you are doing, but a real drag if you make mistakes. This is a down to earth guide of things to avoid.
Do not underestimate the ground. The ground is a surprisingly good conductor of heat, this means any contact with it can really suck the warm out of you. To keep warm you need a big inflatable sleeping pad. Protect yourself because conductive heat loss can not be combated by a tent. Believe me this is worth the investment. I thought I could stay warm when it was 18 below zero with my Therm-A-Rest Prolite 3, big mistake, I had frostbite on my toes where they hung over the end, and I was shivering the entire night. Just bringing a good pad. Also if you can find it, layering lots of leaves or hay under your tent can create and additional barrier between you and the freezing cold earth below.
Put your water bottles inside your bag. I did not think to do this on a different outing and froze my hydration system, which burst it along the side, so not only was my water now uselessless ice, but my Camelback was destroyed. Since then, I have learned, when it is really cold, to only bring Nalgene’s and keep them warm in my sleeping bag over night. Eating snow for water is dangerous, there is no good purification method for it, and it can lower your core temperature and make you really miserable.
Bring a stove you know you can light. On a camp-out when it was -15 I realized that my alcohol stove would not ignite. This meant that I could not turn snow and ice into potable water, not to mention no way to make my dehydrated food. Freezing, hungry, and dehydrated I went to my local gear shop and bought a canister stove, because they will light in any weather. I chose the MSR Pocket Rocket.
Bring a bag made for your conditions. A good sleeping bag is never more important then when it is below zero, I once tried to stretch my 15 degree bag to a simple -5. I am pretty sure I almost died that day. Before the night was through I had filled in my belay parka, 3 beanies, two pairs of pants and my down booties, and still was not warm enough. It was like my bag was not even helping. Lying there, freezing rather than sleeping, I told myself never to make that mistake again. Before buying a new bag, I built a foam insert bag and a down cover to extend the range of my bag, this was a good option in a budget situation, but has nothing on my ultra warm bag.
Get a fourth season tent. That mesh ultralight backpacking tent is great for going on a nice stroll when it is 70 degrees, not even going to help when it is below 0. Make the investment, get a fourth season tent. Black Diamond and The North Face make some excellent options. Because they are built for the wind and the cold, they are predictable in these types of situations.
Remember you can go home. Too many times I have insured that I could survive in the cold, even ill equipped. Lets be honest, no one is going to blame you when you come home and it is -20, but they just might if you get frostbite on your nose, or fingers and toes like I have. Trust me it is not worth saying that you are that tough. Even Shackleton did not want frostbite.