I think in gridview you would have any field like username or ID that you can match with the session of logged in user. To disable the delete right for other you can hide delete button of other rows in your gridview in following way
protected void GridView1_RowDataBound(object sender, GridViewRowEventArgs e)
{
if (e.Row.RowType == DataControlRowType.DataRow)
{
if (!DataBinder.Eval(e.Row.DataItem, "Column_Of_UserName").Equals(Session["User"].ToString()))
{
LinkButton btn = (LinkButton)e.Row.Cells[Indext of your commandfield column].Controls[0];
if (btn != null)
{
btn.Visible = false;
}
}
}
}