The previous answer was generally correct. Having lived in Beijing for 4 years, I can tell you that it can get down to -15C plus wind chill in January. Dull and grey but little snow (maybe 10cm) given that Beijing is in a desert. However, if you like the cold, do not want the crowds and can enjoy the wall with snow, it can be an enjoyable time as you will not be as hassled as normal with tourists. Make sure you find out when Chinese New Year happens in the year you want to go and DO NOT go to China within a fortnight of that week. Everyone is moving around the country. Late March/April can be a good time to visit or September before the National Day holiday on October 1.
Internal flights in China are okay if always crowded as are the internal trains. You can get an overnight train (various classes) from Beijing to Xian then you could fly out from there to HKG or elsewhere in Asia. I would go down south though as well if you have a chance and enjoy the scenery. There is a lot to see in Beijing, not just the Wall so allow some time to explore the city, especially off the beaten track. Arrange to go to the Wall at Mutianyu rather than the much more commercial Badaling - a bit further from Beijing but a much better section of the wall and can still be done in a long day trip.