How to store the value of the textbox in asp.net (csharp ) value in to the javascript array in client side each time when clicking the button .

You could actually use hidden control and assign to the value property of the hidden control each time you click button on client side so that it is persisted across postbacks
var myArray = new Array();
myArray[0] = 'soem value';
document.getElementById('hdnCtrl1').value = myArray;