Integrifolia varieties trail nicely as they are non clinging and often quite small. I have alba and rosea which get to about 3'/1m each year in the ground. They'd need regular feeding and watering to do well in a basket and it would need to be fairly deep for their roots. I grew some in a long conical basket a few years ago but then moved them to a bed.
Species clematis can be grown from seed but for named cultivars you need to do cuttings or cultivars. I found this which explains how better than I could -
Some species start well from cuttings; others do not. The vining types, especially cultivars, are usually started from cuttings or layering. Cuttings should be taken from healthy semi-mature stems of the current year’s wood. Each cutting should have one pair of leaves. The lower end of the cutting can be treated with a rooting hormone and then placed in a medium such as perlite, peat, sand, or a mix of these. Cover the cuttings and containers with plastic to maintain high humidity, then place them in bright light but not direct sun. Once rooted, plants can be potted individually and later placed in the garden.
Many clematis can be propagated by layering (Figure 14). Select 1-year-old stems with good buds. Cut a 1-inch slit in the stem. Pin the stem to the soil in a container or on the ground with a wire. Cover the slit stem with soil and keep moist. Sever the stem when it has developed roots at the slit, gently dig up the rooted stem, and move it to the intended site.
It was on this website - http://www.ianrpubs.unl.edu/epublic/pages/publicationD.jsp?publicationId=1027
The RHS offers this advice on cultivating and propagating clematis - http://apps.rhs.org.uk/advicesearch/profile.aspx?pid=97