Lightroom Sync between two computers is not Possible as of now, but we are working on it and you can already try out the new experimental technology preview Introducing Technology Previews and Search in Lightroom on the web
The other way Meanwhile is to share the catalog by keeping it on an external harddrive. See Sync LR Catalog between 2 computers but not master files
Let us know about your experience with this new feature.
Only real option is to put the Catalog file and all images on an external drive and move the drive from one computer to the other.
Others have used DropBox to sync 2 different computer to the same catalog. But you must have the catalog file, the Real Catalog file, on a drive that is connected directly to each computer.