King of Winter can be quite difficult in general, especially if it is the first World you get. The world is in perpetual winter and you'll only get between 1.5-3 minutes of daylight per 8 minute day. You'll start with a fire blazing on the trees nearby and then after that, you'll have to find your own way of keeping warm. Nearby should be a skeleton with some useful supplies including a Torch, axe, two schematics (one is for Rabbit Earmuffs), a Heatstone and a Divining Rod.
Resources can be quite scarce in this world and may or may not respawn. If you start in a resource rich area, you may want to take your time here and collect as much as you can. You'll want to collect as many Logs, Twigs and Grass as possible when you can. This will be used to make fires as you need them. You'll also want to find some Rabbits and make Traps. Aside from being a good source of food if they are present, they are also very useful for making Earmuffs which will keep you warmer longer.
How much you move to begin with depends on your resources but if you can get a good start on resources, try to keep moving as much as possible. Resources can be quite scarce on this world and may not support you for an excessive period of time. Once you have basic supplies such as Earmuffs, armor and a spear, you should move on. You want to keep an eye out for Beefalo and a Winter Koalefant because killing them can get you some warmer clothing.
To get to certain areas of the map, you may have to pass either Hound Mounds (It will be a stack of bones that Hounds spawn from) or Obelisks on a bridge. The Hound Mound is just a matter of running or fighting your way by (If you fight, try to take out the Mound so the bridge is clear in the future). Obelisks can be tricky. An Obelisk will sink into the ground or raise up depending on the status of your sanity and they will alternate. Some Obelisks require a Sanity above 30 to sink into the ground but some require your Sanity to be below 30 to sink into the ground.
Unfortunately, there's no other way around this except for losing and gaining sanity. If you can find a MacTusk then you may want to take him on to get his Tam O'Shanter which is one of the warmest hats in the game and also provides the best boost to sanity from clothing. If not, hunting spiders can get you some Silk for fancy clothing.
In addition, there's other ways of gaining or losing sanity. Collecting regular flowers, having allies nearby and wearing a Garland will make you feel sane but collecting Evil Flowers, eating raw meat or hopping through Worm Holes will lose your Sanity. The best method, however, is to collect Green Mushrooms. If you eat them raw, you lose sanity but if you eat them cooked, you gain sanity allowing you to draw both qualities from a single type of item.
Being able to hunt to get food and being able to purify Monster Meat through Crock Pot recipes or by using to get Pig Allies is crucial (don't feed pigs more than four pieces or they become Werepigs). This will give you an edge that you might die without.
Again, this isn't an easy world. It requires you to move from one Camp Fire to the next. You'll have to keep a close eye on all of your statistics to survive and even then, the set up of the world just may make it near impossible to succeed anyway so don't be discouraged if you have a few bad runs on King of Winter.