I used 14 gallon pots from Walmart and they are growing very well. To move around I have them on wheels.

Penny

Not sure exactly what your definition of "full size" clematis might be but some are definitely not well-suited to long term container growth. One factor that many folks tend to overlook is that with vines that produce prodigious amounts of top growth, there is an equal amount of growth under the ground necessary to support it, often more. And that takes space to develop. As to daily watering requirements, that too will be dependent on the specific vine selected, the size pot, the potting medium, your climate and how much sun it receives.

In short, there no simple answer to this question, but I'd say Penny's 14 gallon pots are a good place to start (although perhaps not immediately with a young vine) and watering will be as frequently as the plants and your weather demands.