Current Acrobat DC allows user to save local documents to Adobe Document Cloud. User has to have Adobe Cloud services account. You can Sign Up/Sign In from Acrobat DC.
Currently Acrobat DC does NOT support third party cloud services including DropBox. You can email a document to open it and save it into your dropbox.
Are you using Acrobat DC on iOS or Android? On Android, you can open a file from Dropbox in Adobe Acrobat DC by clicking on Open In and then selecting Acrobat DC in the options. After you make any changes to the document, all the changes will be saved back to the file directly. Please let us know if this doesn't help you.
Adobe Acrobat DC Team