We can Set/Get/Clear users’ clipboard by the javascript window.clipboardData object, which only works in Internet Explorer, by the following code:
 
window.clipboardData.setData("text",value)
window.clipboardData.getData("text")
window.clipboardData.clearData("text")
 
If you need set a code-behind value to user’s clipboard, please pass it to javascript variable first, and then, set the value to clipboard. Or you can call the javascript from code-behind, just like the following code:

protected void Button1_Click(object sender, EventArgs e)
{
    string values = "hello world";
    ClientScript.RegisterStartupScript(this.GetType(), "copy", "window.clipboardData.setData('text','"+values+"')", true);
}

Thanks.