How do you separate view from logic in a Winform application?

MVVM(Model-View-ViewModel) Pattern can be used in the Winforms

Model

public class Person
{
    public string FirstName {get; set;}
    public string LastName {get; set;}
}

ViewModel

public class PersonViewModel : INotifyPropertyChanged
{
    private Person _Model;

    public string FirstName
    {
        get { return _Model.FirstName; }
        set(string value)
        {
            _Model.FirstName = value;
            this.NotifyPropertyChanged("FirstName");
            this.NotifyPropertyChanged("FullName"); //Inform View about value changed
        }
    }

    public string LastName
    {
        get { return _Model.LastName; }
        set(string value)
        {
            _Model.LastName = value;
            this.NotifyPropertyChanged("LastName");
            this.NotifyPropertyChanged("FullName");
        }
    }

    //ViewModel can contain property which serves view
    //For example: FullName not necessary in the Model  
    public String FullName
    {
        get { return _Model.FirstName + " " +  _Model.LastName; }
    }

    //Implementing INotifyPropertyChanged
    public event PropertyChangedEventHandler PropertyChanged;

    private void NotifyPropertyChanged(String info)
    {
        if (PropertyChanged != null)
        {
            PropertyChanged(this, new PropertyChangedEventArgs(info));
        }
    }
}

View

public class PersonView: Form
{
    //Add two textbox and one label to the form
    //Add BindingSource control which will handle 
    //ViewModel and Views controls changes


    //As viewmodel you can use any type which of course have same named properties
    public PersonView(Object viewmodel)
    {
        this.InitializeComponents();

        this.ViewModelBindingSource.DataSource = viewmodel;
        this.InitializeDataBindings();
    }

    private void InitializeDataBindings()
    {
        this.TextBoxFirstName.DataBindings.Add("Text", this.ViewModelBindingSource, "FirstName", true);
        this.TextBoxLastName.DataBindings.Add("Text", this.ViewModelBindingSource, "LastName", true);
        this.LabelFullName.DataBindings.Add("Text", this.ViewModelBindingSource, "FullName", true);
    }
}

Read more about databinding in Winforms from MSDN

Obviously WinForms does not natively support one design pattern over another - the one that might not work is MVVM because you cannot "bind" data to the view model and have it update the data directly.

Otherwise - I would attempt WinForms with MVP - I've seen that done before - here's a link to look at https://winformsmvp.codeplex.com/