Which part are you struggling with? (Read the part below and maybe ask specific questions.)
With eagle you setup a path to your libs and it automatically adds all the libs it finds in that path (version 6 knowledge. I have no clue if version 9 functions the same way.)
This gives you 0 control over what you want to have setup.
Kicad on the other hand gives you library tables. These are a bit more work up front but also give you a lot more control over what happens. (you can even give libs a different nickname than their filename. you can choose which lib takes priority, …)
Both are good for different reasons. The two philosophies are not really compatible with each other so the devs need to decide on either one or the other. (Trying to get something in between might not satisfy either side of the userbase)
Your library table would need to be inside of the dropbox folder to get the behavior you want.
Pointing kicad to it can be done using so called symbolic links (created via the right click context menu within your file browser. Sadly the exact wording depends on the language used by you OS.)
The files you need to point to dropbox are found in your users home directory. (windows vista and later: C:\Users\username\AppData\Roaming\kicad, osx: /Library/Preferences/kicad)
The specific files are sym-lib-table and fp-lib-table.
Replace both of them with a symbolic link to the real file that is inside your dropbox folder.
The tables would also need to use path variables instead of using global file system paths. So setup a personal path variable for your personal libs. When you then add your libs using the browse button kicad will automatically use that path variable.
You can get inspiration about that from the official library tables. They use path variables for the reason of being easy to share with users of different operating systems.