How to refresh Form1 from inside a class?

How do I pass "this" from Form1 in the constructor for the class I'm in?

It depends how are you showing your dialog. Here is example solution:

//Main form
    public partial class Form1 : Form
    {
//...
        private void button1_Click(object sender, EventArgs e)
        {
            using(Dialog dlg = new Dialog(this))
            {
                dlg.ShowDialog();
            }
        }
//...
    }


//Dialog form
    public partial class Dialog : Form
    {
        private Form1 form;

        public Dialog(Form1 form)
        {
            InitializeComponent();
            this.form = form;
        }

        private void Dialog_FormClosing(object sender, FormClosingEventArgs e)
        {
            form.Refresh();
        }
    }