When was the picture taken? What variety of clematis is it? In what (US) zone is it growing? By the looks of it the flowering season has finished and what's left is the seed heads. If the flowers have come on 'new' seasons growth, I clip them off and prune back to a pair of strong healthy buds so as to encourage new growth.' more info needed.

When you prune it depends if it blooms on old wood (usually blooming in May) or if it blooms on the new growth for the year, and so blooming in summer. For the summer blooming clematis, some time after hard freeze and before the leaf buds start opening in spring, prune each stem down to 1 or 2 buds per stem. That way the new growth is lower on the plant and will bloom at eye level. If you don't prune them, they may look a bit spindly and only bloom at the top. For the early spring bloomers, wait until after the May bloom to prune for shape and to encourage new growth. Don't prune them low, just remove part of the stems.

Sounds like you're in the US. I'm in Australia and the seasons are reversed (of course). I agree with the advice of NHBabs. Do you know the name of the clematis?

Should I remove the seed fuzzy area and the dead leaves?

When does it bloom?

vettin, before being able to tell you when to prune it people need to know what sort of Clematis it is. Either a cultivar name or information on when it blooms. Pruning is different for different groups of Clematis and they are grouped according to when they flower.

That is most probably an early large-flowered clematis. If you want to remove the old leaves and seed heads you may (they should easily crumple in hand), but I usually find that they fall off when new growth starts if they haven't already.

No need to prune now. This year take note of when it blooms and repost when it blooms for instructions on how and when to prune once we are sure what type it is. Regardless, pruning when unneeded or missing a year's pruning won't hurt the plant overall; it may just not bloom optimally for this year until you know your plant's bloom schedule.