Well, you are going to need 2 things.
1) Something to send the data to WU.
2) Your WU login.
The WU login should be recoverable, if you know the email address you used to register. If you don't know any of that, you may need to start a new page and login.
The first item of the "what to send" with can vary. You may have been using Weatherlink on your old computer. That is the software Davis includes for free.
But there are a couple of other options you may want to look at instead. They don't require a PC. Davis sells an IP logger. A logger is a module that plugs into the back of the weather console. That logger can upload data to WU and Davis's on site. There is also a forum member that sells a logger that uses WIFI. His can also upload to WU and other sites. The only thing I would recommend is that the console use the power adapter. Batteries won't be enough power to last long with either logger. But that will be less power than running a PC 24/7 to upload the data.
If your old PC is still working, you may want to see if there is any software named Weatherlink from Davis, or maybe Virtual Weather Station. If you can find that out, we can probably step you through figuring out your WU login. If the PC is dead, that'll mean you'll likely need to create a new WU login and page.