Observation of the growth cycle for a few seasons is the best way to determine when division should occur.
The best time to divide clematis is just before new growth will appear, and after the plant emerges from dormancy.
When this time is reached, often in late February or March, the plant should be dug up and dirt should be cleared away from the root system. Many clematis plants are climbers, and will be stuck to a fence or other such medium.
It is best to simply cut the plant off, leaving at least three to four good buds per stem.
Once the stems have been trimmed, the jumble of roots should be divided and each plant made should have an even amount of root.
Now, take the separate plants and transplant them into the garden wherever desired. Ensure that the plant will get lots of full sun and water.
It is best to plant the newly divided plants deeply into the soil, to give the weakened roots more protection from drought.