If you're planning on actually PLAYING the games through your computer via a capture card, you're gonna have a bad time. There is a 3-4 second delay, even with devices such as an Elgato Game Cap HD (Which I have previously owned)
Devices like the Elgato have a passthrough to connect your console into it, connect to the computer, and then connect to your television.
Oh and Edit: Just wanted to clarify that COMPONENT is NOT the same as COMPOSITE. If you want to capture the Yellow, Red, and White signals like you have previously stated, that is COMPOSITE. Composite is only capable of 360p/480i. Component is Blue, Green, Red, Red, and White, and is capable of 480p and higher.