It depends on the clematis you choose. I have not grown clematis through bushes (no mature bushes in my yard) but understand that one should choose a shorter variety which will not overwhelm the bush or a nonvining clematis, integrifolia.

Here is one of the threads from the clematis forum that has some info about growing clematis on shrubs. There are others that can be found through the search function on the bottom of the clematis page.

Here is a link that might be useful: How to plant clematis under shrub??

No way would I let anything climb through azaleas - azaleas are pretty fickle, unless you live in Duluth - they need their few leaves to gather as much sun as possible.

Agree with mngal. And the non-vining types of clematis like integrifolia are excellent companions with azaleas and rhododendrons.