You can only play split screen (1-4 players) with the HDMI Cable.
You also have the option of using component cables (red, green, blue, red, white). You cannot play splitscreen if your Xbox 360 is plugged into the television using composite cables(red, white, yellow). If you are using one of the proper cables, make sure the Xbox is set to display in HD. On the Xbox 360, go to Settings>System>Console Settings>Display. It needs to be at least 720p to be able to play splitscreen.
Of course, you need two controllers, but you'll have to purchase an HDMI cable along with a compatible HD TV. You can only play split screen (1-4 players) with the HDMI Cable.
It is a common fallacy that only HDMI cables will work. As Damian4004 mentioned, RGB component cables also work as long as your HD TV is compatible with them and you have the TV and the Xbox properly configured. I played splitscreen for nearly a year using RGB component cables and only converted to the HDMI ones when I wanted to attach another device to my TV and discovered that using any sort of splitter with the RGB compoent cables did not work (and I got quickly tired of manually unplugging one device in order to use the other).
I do believe that the game does run more smoothly with the HDMI cables, but that may just be my impression.
OP: As long as you carefully follow the instructions in the Wiki page Geneo linked, you should have no difficulties getting splitscreen up and running. If you are using RGB component cables and still having an issue, then I would check to make sure that the connection between the Xbox and the TV is direct and not routed through any sort of splitter.