private void button1_Click(object sender, EventArgs e)
{
    new Task(() => RunCommand()).Start();
}

private void RunCommand()
{
    var host = "hostname";
    var username = "username";
    var password = "password";

    using (var client = new SshClient(host, username, password))
    {
        client.Connect();
        // If the command2 depend on an environment modified by command1,
        // execute them like this.
        // If not, use separate CreateCommand calls.
        var cmd = client.CreateCommand("command1; command2");

        var result = cmd.BeginExecute();

        using (var reader =
                   new StreamReader(cmd.OutputStream, Encoding.UTF8, true, 1024, true))
        {
            while (!result.IsCompleted || !reader.EndOfStream)
            {
                string line = reader.ReadLine();
                if (line != null)
                {
                    textBox1.Invoke(
                        (MethodInvoker)(() =>
                            textBox1.AppendText(line + Environment.NewLine)));
                }
            }
        }

        cmd.EndExecute(result);
    }
}