The weather on Mount Kilimanjaro can vary from very hot to extremely cold within the same day although it does not experience wide temperature changes from season to season. You can see a chart of the temperatures on Kilimanjaro below.
What does change a lot is the amount of rain with two distinct rainy seasons: the long rains in April and May and the short rains in late November. If you want to keep dry on your climb avoid these months.
Traffic on the mountain is highest in the drier months but is particularly high during the late European summer months. Flight costs are also generally at their highest during this period.
Also remember 50% of climbers start at the weekend so try to avoid that rush with a midweek start date. And if you want a good clear night to summit look out for the full moon dates.