Make sure the wp-content/uploads folder and its contents are owned by the web server's service account, which if you're using Apache would be 'apache'.
To ensure this, SSH into the server to your web root, and issue this command:
chown -R apache:apache wp-content/uploads
This will change the owner and owner group of all files and folders in wp-content/uploads (including the uploads folder) and should let you upload files through the WP admin.
You may be able to change permissions in your FTP client as well.