You'll first want to make sure all of your cables are connected correctly and in a secure manner. If that's not the issue, make sure your television is set to the correct channel. If those steps do not work, you may need to replace your AV cords.
Many times this is actually just a problem with the AV cables so try replacing those first. If that doesn't work what you are probably looking at is opening the system and cleaning out the dust from the motherboard to assure there is no interference from the video card to the output.
Do you switch your tv to the correct channel to play your xbox? If you don't, then it will turn itself off. Otherwise, call tech support and ask for solutions.
Try the simple things , like buy a different AV cable. If that doesn't work, there is a solution that works most of the time. Get towels to wrap the Xbox in, then turn it on and wait about a half hour, after that, turn it off and wait for about 15 mins. If problem persists, contact Xbox help center.
Make sure that all the cables are connected securely, if that doesn't work then it sounds like you might have a hardware failure. Something the xbox isn't letting the CPU know there is a problem though because you're not getting any Red Lights. Microsoft should fix this problem for free-the only cost would be shipping...