Hi, i want to do speedrun on the xbox 360 but i don't know if it is against the rules to record the screen with a camera or if i am required to use a capture card

as long as your screen is pretty easy to see, you can record it with anything

the avermedia LGP Lite GL310 is a solid starter capture card and it will let you stream your gameplay too. it costs about $50-60 USD

ezcap 284 would be a good capture card to use