What video connection gives the highest quality output on a Wii

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.

Wii systems tend to be less problematic when purchased used, since Wii mods and hardware failures are a bit more rare compared to their install base. It's still possible to get a lemon, or a modified console, so do your research ahead of time.

Wii remotes tend to take a decent amount of abuse as well, owing to the fact that many of the games are aimed at an audience that might tend to be rough on the controllers, and they're intended to be flung around. I've had some bad experiences with used controllers, but I've managed to get good service at GameStop when issues came up.

GameStop publishes their return policy online, which states that used consoles can be returned for your money back within 7 days, and can be exchanged for an identical item within 30 days, if you save your receipt. This is the US policy, and there's a chance that it is enforced differently at your local store, so make sure you confirm this with the local GameStop before you make a purchase.

GameStop can't/won't do an exhaustive check of a system before accepting it for trade; doing so would cause excessive delays for the customer. They'll make sure it turns on and loads a game properly, and that the buttons on the controllers are in working order. Make sure you put it through its paces the first few weeks so that you can confirm it's in good shape before your exchange window runs out.

If you can, you might wait a bit and see if you can get a sale. In the run up towards Christmas, and especially post-Christmas, there are likely to be some good deals.