The weather on Mount Kilimanjaro varies considerably depending on your location on the mountain; land below the mountain is located just south of the equator, so the weather at the gate is warm. As you ascend up the mountain, you will cross five major ecological zones; each zone is marked by a decrease in rainfall, temperature, animal life and vegetation. At night the temperature at Uhuru Peak ranges between -7 and -19 degrees Celsius (20 and -20 degrees Fahrenheit).
Moshi starting point for Kilimanjaro hikes is located south of the equator so the weather is warm and nice most of the year. In Moshi, January and February are the warmest months, April and May are the wettest months, June and July are the coolest months, and August and September are the driest months. The weather patterns in Moshi, generally hold true for Mount Kilimanjaro as well.
Sometimes Mount Kilimanjaro has a mind of its own though; it occasionally creates its own weather(i.e the weather on the mountain can be different from the weather in the surrounding areas down below). Generally, you should expect the weather on the mountain to fluctuate from the one in Moshi sometimes.
The table below shows the average temperature, rainfall, and humidity for Moshi every month of the year.