One possibility is to have a property on the object that is listing to do this:
public virtual int BackgroundId { get; set; }
In the click event you can change the value of this variable to differentiate it from the others:
seuObjeto.BackgroundId = 1;
In XAML, within the datagrid a datatrigger is created:
<DataGrid.CellStyle>
<Style TargetType="DataGridCell">
<Style.Triggers>
<DataTrigger Binding="{Binding BackgroundId}" Value="1">
<Setter Property="Background" Value="Red"/>
</DataTrigger>
<DataTrigger Binding="{Binding BackgroundId}" Value="2">
<Setter Property="Background" Value="Yellow"/>
</DataTrigger>
</Style.Triggers>
</Style>
</DataGrid.CellStyle>
From now on items that have this property changed will also have their background changed. In the example for 1 or 2, but you can set as many as you want in the datatrigger