public class TypingCharAnimationBehavior : Behavior<TextBlock>
{
    private Storyboard _storyboard;
    protected override void OnAttached()
    {
        base.OnAttached();
        this.AssociatedObject.Loaded += AssociatedObject_Loaded; ;
        this.AssociatedObject.Unloaded += AssociatedObject_Unloaded;
        BindingOperations.SetBinding(this, TypingCharAnimationBehavior.InternalTextProperty, new Binding("Tag") { Source = this.AssociatedObject });
    }
    private void AssociatedObject_Unloaded(object sender, RoutedEventArgs e)
    {
        StopEffect();
    }
    private void AssociatedObject_Loaded(object sender, RoutedEventArgs e)
    {
        if (IsEnabled)
            BeginEffect(InternalText);
    }
    protected override void OnDetaching()
    {
        base.OnDetaching();
        this.AssociatedObject.Loaded -= AssociatedObject_Loaded;
        this.AssociatedObject.Unloaded -= AssociatedObject_Unloaded;
        this.ClearValue(TypingCharAnimationBehavior.InternalTextProperty);
        if (_storyboard != null)
        {
            _storyboard.Remove(this.AssociatedObject);
            _storyboard.Children.Clear();
        }
    }
    private string InternalText
    {
        get { return (string)GetValue(InternalTextProperty); }
        set { SetValue(InternalTextProperty, value); }
    }
    private static readonly DependencyProperty InternalTextProperty =
    DependencyProperty.Register("InternalText", typeof(string), typeof(TypingCharAnimationBehavior),
    new PropertyMetadata(OnInternalTextChanged));
    private static void OnInternalTextChanged(DependencyObject d, DependencyPropertyChangedEventArgs e)
    {
        var source = d as TypingCharAnimationBehavior;
        if (source._storyboard != null)
        {
            source._storyboard.Stop(source.AssociatedObject);
            source._storyboard.Children.Clear();
        }
        source.SetEffect(e.NewValue == null ? string.Empty : e.NewValue.ToString());
    }
    public bool IsEnabled
    {
        get { return (bool)GetValue(IsEnabledProperty); }
        set { SetValue(IsEnabledProperty, value); }
    }
    public static readonly DependencyProperty IsEnabledProperty =
        DependencyProperty.Register("IsEnabled", typeof(bool), typeof(TypingCharAnimationBehavior), new PropertyMetadata(true, (d, e) =>
        {
            bool b = (bool)e.NewValue;
            var source = d as TypingCharAnimationBehavior;
            source.SetEffect(source.InternalText);
        }));
    private void SetEffect(string text)
    {
        if (string.IsNullOrEmpty(text) || this.AssociatedObject.IsLoaded == false)
        {
            StopEffect();
            return;
        }
        BeginEffect(text);
    }
    private void StopEffect()
    {
        if (_storyboard != null)
        {
            _storyboard.Stop(this.AssociatedObject);
        }
    }
    private void BeginEffect(string text)
    {
        StopEffect();
        int textLength = text.Length;
        if (textLength < 1  || IsEnabled == false) return;
        if (_storyboard == null)
            _storyboard = new Storyboard();
        double duration = 0.15d;
        StringAnimationUsingKeyFrames frames = new StringAnimationUsingKeyFrames();
        Storyboard.SetTargetProperty(frames, new PropertyPath(TextBlock.TextProperty));
        frames.Duration = TimeSpan.FromSeconds(textLength * duration);
        for(int i=0;i<textLength;i++)
        {
            frames.KeyFrames.Add(new DiscreteStringKeyFrame()
            {
                Value = text.Substring(0,i+1),
                KeyTime = TimeSpan.FromSeconds(i * duration),
            });
        }
        _storyboard.Children.Add(frames);
        _storyboard.Begin(this.AssociatedObject, true);
    }
}

<TextBlock x:Name="source"
            IsEnabled="True"
            Tag="{Binding TypingText, ElementName=self}"
            TextWrapping="Wrap">
    <i:Interaction.Behaviors>
        <local:TypingCharAnimationBehavior IsEnabled="True" />
    </i:Interaction.Behaviors>
</TextBlock>