How do I install a .dmg file without being a sudo user?
.dmg is not an app, it is a disk image. If you double-click the file, it should mount a virtual disk in the Finder. From there, you will be able to drag and drop the image into your applications folder.
Since you don't have admin rights on the computer you're using, you'll need to drag it into your user Applications folder (/Users/yourname/Applications/, needs to be created manually)