When I select a Textfield the keyboard moves over it

Compose an animation and move your TextField container up when a TextField gets focus.

For learning about composing animations refer to: Composing Animations and Chaining Animations in Dart's Flutter Framework

Use Flutter's FocusNode to detect the focus on a TextField

Edit:

Here I've written an example that does exactly what you want:

import 'package:flutter/material.dart';

void main() => runApp(new MyApp());

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return new MaterialApp(
      debugShowCheckedModeBanner: false,
      title: 'Animation Demo',
      theme: new ThemeData(
        primaryColor: new Color(0xFFFF0000),
      ),
      home: new FormDemo(),
    );
  }
}

class FormDemo extends StatefulWidget {
  @override
  _FormDemoState createState() => _FormDemoState();
}

class _FormDemoState extends State<FormDemo> with SingleTickerProviderStateMixin {
  AnimationController _controller;
  Animation _animation;

  FocusNode _focusNode = FocusNode();

  @override
  void initState() {
    super.initState();

    _controller = AnimationController(vsync: this, duration: Duration(milliseconds: 300));
    _animation = Tween(begin: 300.0, end: 50.0).animate(_controller)
    ..addListener(() {
      setState(() {});
    });

    _focusNode.addListener(() {
      if (_focusNode.hasFocus) {
        _controller.forward();
      } else {
        _controller.reverse();
      }
    });
  }

  @override
  void dispose() {
    _controller.dispose();
    _focusNode.dispose();

    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      resizeToAvoidBottomPadding: false, // this avoids the overflow error
      appBar: AppBar(
        title: Text('TextField Animation Demo'),
      ),
      body: new InkWell( // to dismiss the keyboard when the user tabs out of the TextField
        splashColor: Colors.transparent,
        onTap: () {
          FocusScope.of(context).requestFocus(FocusNode());
        },
        child: Container(
          padding: const EdgeInsets.all(20.0),
          child: Column(
            children: <Widget>[
              SizedBox(height: _animation.value),
              TextFormField(
                decoration: InputDecoration(
                  labelText: 'I move!',
                ),
                focusNode: _focusNode,
              )
            ],
          ),
        ),
      ),
    );
  }
}

 Scaffold(
    resizeToAvoidBottomInset: true,
    body: SingleChildScrollView(
      child: Container(
        child: Column(
          children: <Widget>[
            TextFormField(
              decoration: InputDecoration(
                labelText: 'Enter Text',
              ),
            ),TextFormField(
              decoration: InputDecoration(
                labelText: 'Enter Text',
              ),
            ),TextFormField(
              decoration: InputDecoration(
                labelText: 'Enter Text',
              ),
            ),TextFormField(
              decoration: InputDecoration(
                labelText: 'Enter Text',
              ),
            ),TextFormField(
              decoration: InputDecoration(
                labelText: 'Enter Text',
              ),
            ),TextFormField(
              decoration: InputDecoration(
                labelText: 'Enter Text',
              ),
            ),TextFormField(
              decoration: InputDecoration(
                labelText: 'Enter Text',
              ),
            ),
            TextFormField(
              decoration: InputDecoration(
                labelText: 'Enter Text',
              ),
            )
          ],
        ),
      ),
    )
);

// resizeToAvoidBottomPadding: false isDeprecated

use resizeToAvoidBottomInset: true.