Like any plants grown in a container, clematis will need regular fertilization. If it was earlier in the season (spring), I'd suggest a slow release complete fertilizer like Osmocote. But this late in the season. I'd go with a water soluble fert like you would use on any annual container plantings - Dyna Gro or Miracle Gro, etc. - and with every third watering or bi-weekly.

I had a Mandevilla that covered three trellises and had hundreds of flowers on it last Summer. After not fertilizing it for a year, the entire thing turned brown and now looks dead. So figuring out a fertilization schedule is important.

"Flower fertilizers" tend to go low on nitrogen and have more of the phosphorous and potassium. On a flowering vine, too much nitrogen puts the plants energy into leafy green growth rather than flowering. Keep that in mind when selecting a fertilizer.

" On a flowering vine, too much nitrogen puts the plants energy into leafy green growth rather than flowering. Keep that in mind when selecting a fertilizer. "

This is a bit of a gardening myth. Regardless of their life cycle or the production of flowers or fruit, plants will utilize fertilizers in the same ratio - 3-1-2. Osmocote, Dyna Gro and Miracle Gro all have formulations that deliver the necessary NPK nutrients in that ratio. You do not need different fertilizers for different purposes!! That notion is a marketing gimmick rather than a horticultural fact,

btw, lack of fertilization of a containerized plant is highly unlikely to kill it. It just stunts growth and prevents/restricts/limits regular flowering and fruit production.

@gardengal48 (PNW Z8/9) I had posted another thread about using a wholesale bark-based soil that seemed to be robbing nitrogen from plants. I have dozens of plants that all yellow and slowly die that are based on that soil, and they all come back with fertilizer if I catch them before they are totally dead.