The weather in Kilimanjaro is very unpredictable. But it is better to know the common weather of the mountain before you decide to book your tour. The dry season is a good time to climb Mount Kilimanjaro because you will enjoy nature and a good view.  
The dry season is from the beginning of December through the beginning of February. And from late June through the end of October. 
The peak season is from  August to September and the heavy rainy season is from the end of March to the middle of May