Buying a Chromebook doesn't necessarily mean you need to make Google Drive your cloud-storage boo. There is no reason to severe a long and meaningful relationship with Dropbox just because Google Drive is baked into Google's Chrome OS. You can always access Dropbox via the Web on a Chromebook, but a better way is to use a Chrome extension that integrates Dropbox into Chrome OS, putting it right alongside Google Drive.
When you open Chrome OS's file manager (click the Launcher button in the lower-left corner and click Files), you will see two items listed in the left panel: Google Drive and Downloads. These two folders contain your files stored in Google's cloud and locally on your Chromebook, respectively. You can, however, add a Dropbox folder to this left panel to make it easier to access your Dropbox Files.
To do so, you will need to install the File System for Dropbox extension from the Chrome Web Store. The extension isn't from Dropbox but a Japanese developer by the name of Yoichiro Tanaka.
After installing the extension and granting it access to your Dropbox account, it will mount Dropbox to Chrome OS and add a Dropbox folder to the left panel of your Chromebook's file manager. With Dropbox added to the file manager, it's easy to drag and drop files from the Downloads and Google Drive folders to Dropbox, and vice versa.
The File System for Dropbox extension provides access to all of your Dropbox folders, including folders others have shared with you. It does not offer offline access to your Dropbox files, but it stays mounted through a shutdown and restart. Should Dropbox go missing from your file manager, you can always remount it by finding the extension among your apps via the Launcher button. (It's simply labeled Dropbox with a navy blue folder icon.)
A similar extension from the same developer is available for those who use Microsoft's cloud service; check out File System for OneDrive.