First, since you need to access an existing shared folder, register a “Dropbox API” app with “Full Dropbox” access:
https://www.dropbox.com/developers/apps/create
Then, get an access token for your account for your app. The easiest way is to use the “Generate” button on your app’s page, where you’ll be sent after you create the app. It’s also accessible via the App Console.
Then, you can upload to a specified path via curl as shown in this example:
This uploads a file from the local path matrices.txt in the current folder to /Homework/math/Matrices.txt in the Dropbox account, and returns the metadata for the uploaded file:
echo "some content here" > matrices.txt
curl -X POST https://content.dropboxapi.com/2/files/upload \
--header "Authorization: Bearer <ACCESS_TOKEN>" \
--header "Dropbox-API-Arg: {\"path\": \"/Homework/math/Matrices.txt\"}" \
--header "Content-Type: application/octet-stream" \
--data-binary @matrices.txt
<ACCESS_TOKEN> should be replaced with the OAuth 2 access token.