Failure to Prepare is preparing to fail
Make sure you study your chosen route carefully and make sure you are aware of any dangers that may be present. These can include ridges that may need to be crossed or scrambles to be climbed, these areas can be dangerous enough in normal conditions, but in winter when there is snow and ice they can be deadly.
Make sure you have the correct equipment for the hike. In wintery conditions the equipment needed is usually quite different that the other seasons. You may need crampons and an ice axe if you are going somewhere with heavy snow of sheet ice. If you do need this equipment make sure you are comfortable using it first as these tools can be dangerous if you are unprepared. There are usually lots of courses available or guides to help you if you need to learn how to use them.
Navigation equipment- you will need a GPS and/or compass and know how to read them, the weather can turn quickly and you will need to be able to make sure you know where you are and where you are going if the route becomes obscured. Make sure you have maps with you of the chosen route and are familiar with them before you set out.
Comfort and safety- Make sure you take adequate clothing with you and good footwear. For hill walking always make sure you have several layers of clothing, this makes it easier to adjust as you hike. When walking you will often feel warmer than when you stop for a break so having an extra warm layer to put on is always useful. Also be aware that the higher you go the colder it will be so even if you feel warm when you set out at a low altitude chances are that when you are nearer the top it will be much colder.
For walking in winter make sure you have good sturdy boots. You will want them to have ankle support and be waterproof. If they are new take a few low level walks in them first to break them in and make sure they are comfortable, you don't want any unexpected surprises when you are up a hill.
It is also advised to take a small emergency shelter and a foil blanket, these don't weigh much but can be the difference between life and death in a emergency situation. There is a wide variety of these available on the market from simple one man bags to ones that can hold a small group. Make sure you have an adequate one the situation.
Other things you will need is adequate water and food. Make sure you know how long the route will take you and plan how many supplies you think you will need. Remember whilst walking you will be using much more energy than usual so take plenty of snacks such as trail mix or energy bars.
Make sure you have the phone numbers for the local emergency services and mountain rescue teams, these can be invaluable if anything goes wrong. Also have someone who knows where you are and your route plan. If anything goes wrong they can send for help and make sure you are safe.
Finally take care, be prepared and have fun.