The best time is probably never !!!!
They hate root disturbance and often die off when moved.

If you HAVE to though it's probably best to do it in the winter when it's presumably dormant - if it's a winter flowerer that clearly won't apply though !

Without knowing much about the type it's a bit difficult to say - perhaps someone else will have a better idea.

Tend to agree with Louise they have a deep root run and hate being disturbed.

I planted next to a clem once, just tickled its roots, and it died back immediately - they really don't like their fibrous roots being moved !

Hmm, strange, I've moved clematis 3 times - the same one twice, and another one once, without any problem. Both times at this time of year, but they were summer flowering ones...