How can I upload a PDF using Dart's HttpClient?

You can use MultipartRequest from the http package :

var uri = Uri.parse("http://pub.dartlang.org/packages/create");
var request = new http.MultipartRequest("POST", url);
request.fields['user'] = '[email protected]';
request.files.add(new http.MultipartFile.fromFile(
    'package',
    new File('build/package.tar.gz'),
    contentType: new ContentType('application', 'x-tar'));
request.send().then((response) {
  if (response.statusCode == 200) print("Uploaded!");
});