Within Teamwork, you can upload files directly to your integrated storage apps such as Dropbox and Google Drive.
In the Files area of your project, click the File Integrations button in the top right to select the storage integration you want to upload to.
In the integration modal, you will have an option to upload in the top right corner.
Once you click Upload, you will be prompted to choose a file and it will be added to your Dropbox/Google Drive/OneDrive/Box.com folder. From the same screen, you can also then link the file directly to the project.
For more information, see: Linking Files from Integrated Storage Accounts to Your Project