That would be 'Scartho Gem' and a very nice choice for a container. I typically use a 3 gallon container as a minimum size for any of my containerized clematis but ultimate size depends heaviy on the size of the vine. A 15" diameter pot is pretty close to a 3 gallon capacity so I'm not sure I'd go much larger.

Container growing does have some requirements that inground growing does not. You need a very good quality potting soil (not something like Miracle Gro) and you need to pay very close attention to both watering and fertilizing. You should also repot relatively frequently - even high quality potting soils break down over a season or two and need to be replaced. And your vine should be root pruned at that time as well.

Thanks gardengal for the good advice...I think I will repot this year after summer. It has blooms coming already! This was last year: