You need to clear an area so that there is little snow on the ground; eventually, your fire will settle down to the ground, so, the less snow there is underneath, the better.
In really bad situations, you can't move the snow (maybe it's icy, or you don't have a shovel or easy means to move the snow?) or the depth isn't known. What you can do is to lay larger logs to make a flooring of sorts, then build your fire atop that.
Another technique is to build a fire on top of a rock; again, you will have to clear snow, but if that's not feasible, you could still lay down flooring logs.
You might also want to build a fire on top of a tree stump. You might find some debris, like a pot or a #10 tin can in which you can build your fire. Or large wet strips of tree bark. Or a fallen tree. Or maybe you have aluminum foil in your gear; use that for underneath the fire.
Keep in mind that a survival fire should have two purposes: to keep you warm, and to make you noticed. In keeping you warm, you need to locate it so that it won't melt snow in your shelter area; that means clearing out as much snow as possible around your shelter area. Again, make a log flooring if this isn't feasible.
To make you visible, build the fire at higher elevations, so that any smoke generated won't blend with any trees making it harder to pinpoint you.
Starting the fire
You didn't indicate your experience in starting fires, so, for posterity, I'll just assume you don't know; if anything, the tips can be useful for others.
If you don't have firestarters, you can use birch bark—and only birch bark. Such can be found by its distinctive white (or silver) coloring, and which can occasionally be found peeling from a tree or log. You want to collect it from ground logs to prevent injury to a living tree; however, in survival in the snow, it may be impossible to find on the ground: you'll have to get it from a live tree. Birch bark contains a waxy substance which makes it nearly waterproof and easily lit. That is your firestarter (hopefully, you have matches or a lighter?). But take only what you need, and no more, unless you're getting it from dead trees/logs.
In areas where birch isn't available, an excellent firestarter is Old Man's Beard. It grows everywhere, and is most commonly found on pine and oak branches, sometimes hanging in long, beard-like strands. It is a light grey-green color, and it will light even when wet.
Use the fire starter to light small wood -- tinder -- that easily snaps when bent. This should be dead wood, not green from a tree. In snow conditions, it's freezing, and hard to tell if the wood is dead or just frozen if you don't have experience. If the twigs come from a branch with leaves on it, it's probably green or live, so keep looking. Green tinder won't light well.
Use the tinder twigs as a base to light slightly larger wood, maybe 1" in diameter or less. Build this wood in a neat fashion, like a tee-pee shape, or a lean-to shape. It should be built to shield the tinder from being blown out from any wind, snow, or rain.
Lay larger pieces on top of the small wood. Don't crowd the fire, give it a chance to catch and breathe. Once going, you can add larger wood.
In a survival situation, you need to keep warm. If you have a reflecting blanket/tarp, wrap it around the coldest member of your group, with the shiny side inside. This causes the dark side to absorb sunlight, the exact opposite of what you want to do in scorching heat. Everyone should keep moving to keep the body core as warm as possible: this will help to ward off hypothermia.
One way to keep moving is to keep looking for firewood. You don't know for how long you'll need to be there. Maybe you're lost, or maybe you're not but you're stuck in a weather pattern preventing you from moving. Whatever the case, have ample firewood on hand. Anyone suspected of having hypothermia should not be left unattended: they may wander, creating a new survival problem for you. But they should be moving as much as possible.
Eat, drink, and be merry
Once your fire is built and you have a shelter in case you'll be there awhile, next up is to find water, food, and help, pretty much in that order. It is most important to have a good attitude. That will help you think clearly and be proactive about what to do. When/if hypothermia does set in, a depressed attitude will be more noticeable, so, there are many reasons for having a positive and upbeat attitude, including from those who are normally sullen. Being depressed in cases like this creates arguments, makes it harder to identify hypothermia, and generally is a selfish attitude. This is not the time to be selfish. Water is easy, just melt or eat snow. (That's a good way to create other medical problems, but worry about that later). Do eat snow sparingly, though it will cool the core -- the very thing you don't want to do right now. Next up is food. You may have to do without, unless you're good at hunting, trapping, or fishing.
Calling for help can pose challenges, too. You might have a cellphone, use that. Yelling is a waste of time, unless you're near a well-traveled trail. Anyone hearing you won't be able to pinpoint you, and snow will absorb sound, so, you won't have benefit there. A whistle can help, and will save your voice if you need it.
Use a reflector -- like a mirror -- to reflect sunlight on passersby you might see, or on buildings you see in the distance.
The fire building technique doesn't change regardless of the snow on the ground. What changes is the amount of work you have to do to get the snow out of the way, or why you are in survival mode at all. If you have only a light dusting of snow, or you've cleared most -- but not all -- of it away, you might decide that you have enough firewood to not worry about melting snow putting your fire out.