You just download and install the version you have paid for. You install the same way as you did the first time. The two computers cannot share a catalog. Each computer must have its own catalog. Or, you complete your catalog and images on an external drive to switch between computers.
Personally I have my catalog and previews on a LaCie Thunderbolt drive and image files on various USB drives. Lightroom I have installed on my two MacBooks.
With this set up I can work either laptop, the only thing to do is ensure synchronising of presets, etc.