There are a lot of options.
First off, you will need the Davis WeatherLink package. That includes Davis's software and a data logger that plugs into your Vue console and then to your computer. There are USB and RS-232 versions of the WeatherLink package. I highly recommend the RS-232 (serial) version -- the USB version gave me all sorts of problems. If your computer doesn't have an RS-232 port (mine does not), a good USB<->RS-232 adapter will work.
You can use the Davis software to create web pages and upload them to a server. The software is rather "old school" in look and feel, but it does work. It does not provide "real-time" updating, though, to your own website. I believe it will do "rapid fire" updates to weather underground if you've set up an account there, though. There is a software "plug in" that you need to get from Weather Underground to enable this in WeatherLink.
I use a program called WeatherDisplay for gathering data from my Davis and sending that info to my website. It's not free, and there's a steep learning curve, but it has the most features. A free program that works well is called Cumulus -- worth looking into. But again, the Davis software that you're forced to buy along with the (necessary) data logger will work, too.
You can see my personal weather website here: http://www.wrightwoodweather.com
And you can see my Weather Underground page here: http://www.wunderground.com/cgi-bin/findweather/getForecast?query=92397&sp=KCAWRIGH6
Finally, you can get all the info and help you can imagine here: http://www.wxforum.net/