What to expect during the winter season
Winter in the Rocky Mountains generally begins around mid-October and at the higher elevations it can last through April or early May. While most people think of fresh fluffy snow when they think of winter in the Rocky Mountains the reality for the eastern side of Rocky Mountain National Park is that winter here is usually characterized by high winds. For most of the winter the winds blow quite strongly, often gusting over 50mph. Above tree line the winds are even stronger. Plan your days around the wind conditions and avoid hiking in the forests when the winds are strong, as dead trees often fall during these strong winds.
October through January tends to be fairly dry. While there is likely to be some snow, the largest snowfalls tend to arrive in late-January through early-May. Most of the snow will be found in the subalpine zone (9,000-11,500') which catches the snow blown off of the tundra.
At this link you'll find a description of what to expect for each month of the year.
While the overall size of Rocky Mountain National Park may be relatively small, the weather can vary drastically from place to place, especially as you change altitude. It is therefore wise to look at several forecasts to get an idea of what to expect. Be sure to not only look at the temperature and the chance of storms, but also the expected wind speed as that can impact how cold it feels. The winds can at times be so strong (+40mph) as to make hiking dangerous and topple trees. While these forecasts by the National Weather Service are very helpful, it is important to note that accurately predicting mountain weather is extremely difficult, so always prepare for unexpected changes in the weather.