The two main trekking seasons on Mount Kilimanjaro are also the warmer / dry seasons on the mountain, Mid December to mid-March and June to October. From January to mid March skies should be clear and temperatures warm, but a little cooler than from June to October. Trekkers can expect the odd rain shower in the afternoons.
Early June, after the rainy season there may still be the chance of cloud cover, but generally conditions should be dry and clear throughout this period, becoming increasingly warmer. This is peak Kilimanjaro climb season due to holiday periods in the northern hemisphere and so trails may be busier. Inclement weather can occur at any time however, and so trekkers should be prepared with the appropriate clothing.