Uhh, hi there

The Gamecube comes with A/V cables only. To hook these to a VGA monitor, you'd need a "VGA box". I believe that Redant makes a popular model for under 50 bucks (I could be wrong about the price). Picture quality won't be so good.
The next step up the quality ladder would be S-Video-to-VGA. You'd need to purchase a Gamecube S-Video cable and a box to convert S-Video to VGA. I don't know the specifics of such a device, but I'm sure they exist. Quality will be better, but still not true VGA quality. This is probably a little more expensive than the previous option.
Finally, you get into the realm of Component Video to VGA. This is where the quality is, and this the option with which I am particularly familliar. It can be cheap, but you probably don't want to go this route, since you'll only be able to play progressive scan games unless you have a really rare monitor. There are really three ways to go about it.
1) Buy a component video cable from Nintendo and mod it yourself to produce a VGA signal. Cheapest option, but requires soldering sk1llz.

2) Buy a pre-modded cable from somewhere. (Goldenshop, I think)
3) Buy a component-to-RGB transcoder. This is expensive.
If I remember correctly, BNC connectors carry the same signal as a regular 15-pin VGA, it's just a different physical connector. It doesn't really help, unless you mod your own cable in which case it may save you a little soldering.
Unless you are a tinkerer, or a real entusiast, or just have money to burn, you probably want to go with one of the first two options.
-Angus