Mt Kilimanjaro is caused by a divergent movement, the two parts of a plate are being pulled away from each other. It does not actually sit on a boundary, it sits on the African Plate, which is fractured. The two sides are moving away from each other, causing the divergent movement.
The fracture was caused by convention currents found in the Earth's mantle, which lies just beneath the crust. The land in between the fracture dropped down and caused a deep valley. This is known as the Great Rift Valley (sometimes the African Rift).
So it is actually the same plate splitting that caused the volcano, rather than a plate boundary. In the valley there are open fissures and this is where parts of the mantle can escape back up to the crust.
Here you can see where the rift is on the African Plate, it runs right down the centre and is not near to a boundary at all. Mt Kilimanjaro lies within this rift.
The volcano and the rift were created at the same time, about 3 million years ago, along with a lot of other volcanoes.
At first Kilimanjaro was just a region, and there was lots of volcanic activity centred within the region. However over time all the activity began to focus on just three points, which make the peaks of Mt Kilimanjaro today. Their names are Shira, Mawenzi & Kibo.
It is now the largest free standing mountain in the world.