The mountain can be climbed all year round. The main climbing seasons on Kilimanjaro correspond with the mountain's two 'dry' seasons. These are January to mid-March and June to October. Note that it will still probably rain on your climb during these months too - it's rare to climb without getting rained on at least once. Know that you can even take a climb in the rainy season but there is a much higher chance you'll encounter rain, and the summits of Kibo and Mawenzi are likely to be wreathed in the thick cloud too. Indeed, several agencies even suspend their operations in November and December.
It's true that visibility is less at this time and even your chances of reaching the summit are significantly lower too; that simply is not the case. It is true, once every couple of years or so there may be too much snow, blocking you from reaching the summit. But this is rare. So as long as you are dressed for the conditions then, in our experience, you have got just as good a chance of getting to the summit.
Curiously, Christmas and New Year, when the weather is far from perfect, are actually amongst the most popular times for climbing.