Catalogs from different versions are not compatible. You could copy your Lightroom 4 catalog to the new computer and open it using Lightroom 5. That will initiate a process of creating a new copy of that catalog in the Lightroom 5 format. But be aware that converting the catalog will not automatically bring the images. Images are not in the catalog. It would also be necessary for you to copy the images to the new computer as well.
Also note that images are not interchangeable between the two versions of Lightroom. Catalogs are version-specific. You probably should update Lightroom on the older computer if the specs will allow it.