No this will not help with a console's fps. An Xbox One/PS4 has a max fps of 60fps and an Xbox 360/PS3 usually get around 30 fps, you cannot change this because you cannot upgrade a console like a PC.
A capture card is strictly for recording game plays for social media (eg. Youtube) or for personal use. But if you're into that sort of think I would recommend buying one, I have one but no longer use it.