my StreamReader and Split Method doesn't worked. O.o

To cycle through the records, you'll have to save them all to an array first and so the code is going to need to be changed quite a bit. Something like this (off top of my head so untested):

using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Drawing;
using System.Text;
using System.Windows.Forms;
using System.IO;

// add private fields
private string[] lines;
private int nextLine = 0;

        private void frmCustomerDetails_Load(object sender, EventArgs e)
        {
            // get all lines from file and store them in 'lines' array
            lines = File.ReadAllLines(@"C:\mp2.txt");
            // display first one
            DisplayRecord(0);

        }

        private void DisplayRecord(int index)
        {
                string[] records = lines[index].Split('#');
                txtCustomerID.Text = records[0];
                txtContactPerson.Text = records[1];
                txtAddress.Text = records[2];
                txtContactPerson.Text = records[3];
                txtContactNo.Text = records[4];
                if (index < lines.Count - 1) nextLine = index + 1; //  change nextLine
        }

        // add these eventhandlers in designer so they're automatically wired up
        private void btnFirst_Click( object sender, EventArgs e)
        {
             DisplayRecord(0);
        }

        private void btnNext_Click( object sender, EventArgs e)
        {
              DisplayRecord(nextLine);   
        }

Thank's a lot. That really worked.

Suppose I have three lines of records on my mp.txt:

123#A#qwe#Z#321#

456#B#asd#Y#654#

789#C#zxc#X#987#

And I have this buttons named ' btnFirst' and 'btnNext'

wherein btnFirst - displays the first list

btnNext - displays the next after the first

How can I able to put that in codes in the btnFirst and btnNext, along with the previous corrrect codes?

Try changing your while statement to this:

            while (!reader.EndOfStream)
            {
                line = reader.ReadLine();
                records = line.Split('#'); // single quotes here

                // all these were the wrong way around!
                // no need for any conversions - they're all strings already 
                txtCustomerID.Text = records[0];
                txtContactPerson.Text = records[1];
                txtAddress.Text = records[2];
                txtContactPerson.Text = records[3];
                txtContactNo.Text = records[4] ;
            }