Clematis can definitely be grown in a hanging basket if you select the right variety and then give it a little training.
Be sure to use a minimum 30-centimetre basket. Many varieties of clematis will work, but the key to finding one that will bloom well in a basket is to choose one that will bloom on the current season's growth or, in other words, a summer bloomer. Clematis 'Ville de Lyon' and C. tangutica 'Golden Harvest' are two examples of the many varieties that will work.
As most gardeners know, clematis grow very rapidly until they set bud. This growth should be wrapped around the wire of the hanging basket until the flower buds begin to form. At that point the training is finished, and any new growth can be allowed to trail over the sides of the basket.
Your basket should then continue to bloom through the summer and into early fall. For the winter, move the basket to a protected spot outdoors. When the plant is dormant prune it hard, and in spring hang it up again and enjoy its beauty for another season following the above procedure. After one or two seasons in the original basket, transplant the clematis into a larger basket or in the garden. Related link: Video on pruning clematis.