The climbing/trekking season is year round, but there are two rainy periods, mid March to mid June and mid October to mid December. The rainfall is less the higher you go, but it can rain or snow any day of the year.
Mt. Kenya lies on the Equator, so the climbs on the south side get more sunshine in the December to March season and those the north side more sun from June to October.
Wind chill rating: Medium
Most people approaching the mountain will have the use of porters, so the weight of the gear will not have such a direct impact. If you are carrying all your own kit, then all the comments on lightweight will apply.
Looking for a guided trip with an experienced company? We recommend KE Adventure Travel's Mount Kenya Ascent.