How to delete a selected DataGridViewRow and update a connected database table?

I have a DataGridView control on a Windows Forms application (written with C#).

What I need is: when a user selects a DataGridViewRow, and then clicks on a 'Delete' button, the row should be deleted and next, the database needs to be updated using table adapters.

This is what I have so far:

private void btnDelete_Click(object sender, EventArgs e)
{
    if (this.dataGridView1.SelectedRows.Count > 0)
    {
        dataGridView1.Rows.RemoveAt(this.dataGridView1.SelectedRows[0].Index);
    }                
}

Furthermore, this only deletes one row. I would like it where the user can select multiple rows.