This is the DataGridView handler you need to capture the button click event. 
this.dgvList.CellContentClick += new DataGridViewCellEventHandler(DGV_CellContentClick);
This is the button click handler example 
#region DGV_CellContentClick
public void DGV_CellContentClick(object sender, DataGridViewCellEventArgs e)
{
    int selectedRowIndex = int.Parse(e.RowIndex.ToString());
 
    if (this.dgvList.Columns[e.ColumnIndex] == buttonColumn && selectedRowIndex >= 0)
    {
        DataRow dr = DataGridViewHelper.GetDataRow(this.dgvList);
        MessageBox.Show((string)dr["FirstName"]);
    }
}
#endregion