You want to buy the component cable, which supports the Wii's maximum output resolution of 480p.
480p (DVD-quality) looks much better than the 480i (more like pre-DVD, analog TV quality) you get with the (included) composite cable.
It's still not the 720p, 1080i, or 1080p that newer TVs can handle, but the Wii doesn't output in those resolutions.
Probably the cheapest way to do it is to use a USB capture card. Personally I use the EzCap116 which works nicely.
You can hook the cable and device directly into your computer and play using capturing software. I recommend AmaRecTV, which will allow you to record video, play with aspect ratios, deinterlace, full screen and so on.
If you want to play on your TV while recording on your computer, you'll want to split the signal. I use a stackable composite splitter similar to the ones sold here on eBay:
If you want to see an example of this setup, here's a video of somebody connecting their Xbox 360 in this manner; it will work the same for the Wii.
EzCap also has their own official YouTube channel, with a setup video, which uses 3x female-to-female phono splitters. I recommend doing this so your video quality stays consistent between the displays you are splitting to.
A word of warning though: products named "EzCap" are made by an official company, products named "EasyCap" are bootlegs, as stated on their drivers FAQ.
You may also want to check out EzCap's official website.
Follow up: I have been using this setup for a while, mostly for Wii videos myself. I wanted to comment on the stackable composite splitter that I previously recommended.
The splitter I recommend above is not a standard cable. You will not find it anywhere in standard retail outlets -- only specialty shops and bootleg type cables.
The reason for that is because this specific cable suffers from a severe loss of quality on at least one end. The colors are dark and washed out. I do not recommend this cable if quality on both displays is something you are concerned with. If this is the case, go with something else, such as the separate splitters mentioned above, or get a switch box that handles this.
The phono splitters are also dirt cheap.
I regret buying the cable. I will say, however, that it is very convenient to only have one piece of hardware that splits your composite signal into two. But it is not worth the loss in quality nor the price.
Best Answer
Your Wii is probably set to PAL 50, which causes would cause similar problems. Setting the Wii to PAL 60 would work, http://www.nintendo.co.uk/NOE/en_GB/support/wii_support_5251_8159.html shows how to change that setting, under "Screen".
It's a "bit" of a pain when the screen looks like that but it's doable if you keep the wiimote in the same position when you can't see the screen. (My TV shows a few frames before a few seconds of and then repeats, don't know if your tuner does)