Basically, you are going to definitely need more hardware to do this unless you have a monitor with built in composite video ports (the red white and yellow ones).
You could do either of the following:
If you want to hook it straight into an additional port in the monitor (say you have more than one RGB/DVI port) the you will have to buy a converter box simliar to this one:
http://www.howtoconvert.co.uk/howcomp2rgb.htm
If you have a desktop, you should buy a TV tuner card that has a composite video in. If you go this route, you will have to install media center software, such as a MythTV if you run linux or Windows MCE. Some tuner cards come with software to do this on top of Windows as well, which is often the easiest route but provides limited functionality. Also, when purchasing a TV tuner card, your quality will greatly depend on which card you get. To have crisp Wii graphics on your monitor, you will need to get an HDTV card as well as HD cables for your Wii.
Last, if you have a laptop, you will need to get a USB TV tuner card. These cards do the video conversion via software, and are definitely your worst option.
In conclusion- you probably would just be better off buying another TV unless you want to go ahead and make a media center out of your computer. I made a MythTV using Ubuntu server a while back, and it is definitely worth it if you have an extra machine, a little cash, and the time : )
