<Window.Resources>
    <BooleanToVisibilityConverter x:Key="VisibilityConverter" />
</Window.Resources>

<StackPanel>
    <Grid>
        <TextBox
            Text="{Binding Password, UpdateSourceTrigger=PropertyChanged}"
            />
        <PasswordBox
            x:Name="PasswordBox"
            PasswordChanged="OnPasswordChanged"
            Visibility="{Binding HidePassword, Converter={StaticResource VisibilityConverter}}"
            />
    </Grid>
    <CheckBox
        Content="Show password"
        IsChecked="{Binding ShowPassword}"
        />
</StackPanel>

public class ViewModel : ViewModelBase
{
    private string _password;
    public string Password
    {
        get => _password;
        set => Set(ref _password, value);
    }

    private bool _showPassword;
    public bool ShowPassword
    {
        get => _showPassword;
        set
        {
            Set(ref _showPassword, value);
            RaisePropertyChanged(nameof(HidePassword));
        }
    }

    public bool HidePassword => !ShowPassword;
}

public partial class MainWindow : Window
{
    public MainWindow()
    {
        InitializeComponent();
        InitializeViewModel();
    }

    public ViewModel ViewModel => DataContext as ViewModel;

    private void InitializeViewModel()
    {
        DataContext = new ViewModel();

        ViewModel.PropertyChanged += (sender, args) =>
        {
            // Update the password box only when it's not visible;
            // otherwise, the cursor goes to the beginning on each keystroke
            if (!PasswordBox.IsVisible)
            {
                if (args.PropertyName == nameof(ViewModel.Password))
                    PasswordBox.Password = ViewModel.Password;
            }
        };
    }

    private void OnPasswordChanged(object sender, RoutedEventArgs e)
    {
        ViewModel.Password = PasswordBox.Password;
    }
}