You can connect your console to a monitor provided it has HDMI.
You can't connect your console to your PC though. A capture card wouldn't really work, as it's really just mean to capture a recording of the console gameplay, you're not really playing the game on the PC at all, you're just watching a recording of yourself playing the game in real time with a slight delay probably.
If you want to record your gameplay for youtube, than yeah a capture card is what you want, but you'd still play it on your tv/monitor like normal and then edit/upload the video file later.