It is possible all year long although most avoid the rainy season: The long rainy season is from April to May, the short rainy season is mostly in November/beginning of December. Always be prepared for wet days and cold nights on the mountain any time of year!
Each month has its specific (dis)advantages:
Jan-March: The warmest months: few clouds - one of the best times to climb.·
April- May: Rainy season, many clouds, rain on the lower slopes, maybe some snow on top. Still warm though and the least crowded months.
June-July: Clear skies, cold nights, beautiful!
Aug-October: Warmer, busy months; October is least busy of this high season.
Nov-December: Short rainy season, (clearing) clouds and possible thunder; Snow on the higher slopes. End of December (white X-mas!) is getting clearer and warmer, very beautiful
By the way we organize our trips all year round, so anytime you wish to climb we can arrange it. Some people actually prefer to climb during the rainy/snowy season (which can be quite dry actually) so they can see fresh snow on Kilimanjaro and have the mountain to themselves as it is much more quiet on the mountain then! Worth to think about if you are not afraid of a little winter in Africa :-)