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playername = Convert.ToString(PlayerNameEingabe.Text);

Player player1 = new Player
{
    PlayerName = playername,
    PlayerID = playerid,
    PlayerScore = 0,
};
DataGird.Items.Add(player1);
playerid += 1;					

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<DataGridTextColumn Binding="{Binding PlayerID}" ClipboardContentBinding="{x:Null}" Header="ID" HeaderStringFormat="PlayerID"/>
<DataGridTextColumn Binding="{Binding PlayerName}" ClipboardContentBinding="{x:Null}" Header="Spieler Name"/>
<DataGridTextColumn Binding="{Binding PlayerScore}" ClipboardContentBinding="{x:Null}" Header="Punkte"/>					

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// als Member
private ObservableCollection<Player> players = new ObservableCollection<Player>();

// im Konstruktor
DataGrid.ItemsSource = players;

// im Button Click
Player player = new Player { /*...*/ };
players.Add(player);