Form1.cs

using System;
using System.Windows.Forms;

namespace Prueba_listview
{
    public partial class Form1 : Form
    {
        public Form1()
        {
            InitializeComponent();
        }
        public void listar()
        {
            string [] nombres = {"fernando","ignacio","mirta","florencia" };
            string[] apellidos = { "campos", "gomez", "alonso", "farinelli" };
            for(int num=0;num<=3;num++)
            {
                ListViewItem item = new ListViewItem();
                item.Text = num.ToString();
                listView1.Items.Add(item);
                listView1.Items[num].SubItems.Add(nombres[num]);
                listView1.Items[num].SubItems.Add(apellidos[num]);
            }

        }
        private void listView1_ColumnClick(object sender, ColumnClickEventArgs e)
        {
            listView1.ListViewItemSorter = new ListViewItemComparer(1, SortOrder.Ascending);

        }

        private void button2_Click(object sender, EventArgs e)
        {
            listar();
        }
    }
}

ListViewItemComparer.cs

using System;
using System.Windows.Forms;
namespace Prueba_listview
{
    public class ListViewItemComparer : System.Collections.IComparer
    {
        private int col;
        private SortOrder sortorder;
        public ListViewItemComparer()
        {
            col = 0;
        }
        public ListViewItemComparer(int column, SortOrder sortOder)
        {
            col = column;
            sortorder = sortOder;
        }
        public int Compare(object x, object y)
        {
            int resultado = String.Compare(((ListViewItem)x).SubItems[col].Text, ((ListViewItem)y).SubItems[col].Text);
            if (sortorder == SortOrder.Ascending)
            {
                return resultado;
            }
            else
            {
                return resultado * -1;
            }
        }
    }
}