GraphicsLayer="{Binding Path=Layers[Parcels], ElementName=myMap}"
Also, FeatureDataGrid automatically populates columns for you unless you explicitly set its AutoGenerateColumns property to FALSE. I suggest do the above with the default behavior (AutoGenerateColumns = true) and see if it works then try to set your own column collections with their corresponding alias headers.
<esri:FeatureLayer ID="Parcels" OutFields="*" Url="http://sql2k5/ArcGIS/rest/services/CityBasic/MapServer/8" FeatureSymbol="{StaticResource SelectSymbol}" />
<esriToolkit:FeatureDataGrid x:Name="featureGrid1" Map="{Binding ElementName=myMap}" GraphicsLayer="{Binding Path=Layers[Parcels], ElementName=myMap}" />
FeatureDataGrid should be populated by the features you have in your layer as it takes care of the CollectionChanged property of your layer's Graphics object.
xmlns:esri="http://schemas.esri.com/arcgis/client/2009">
<Grid x:Name="LayoutRoot" Background="White">
<Grid.Resources>
<esri:Editor x:Key="MyEditor" Map="{Binding ElementName=MyMap}" SelectionMode="Rectangle" />
</Grid.Resources>
<esri:Map x:Name="MyMap" Extent="-102.049995692435, 36.995429186312, -94.6031396158311, 40.0025826914111">
<esri:ArcGISTiledMapServiceLayer ID="Tiled" Url="http://services.arcgisonline.com/ArcGIS/rest/services/ESRI_StreetMap_World_2D/MapServer"/>
<esri:FeatureLayer ID="Parcel" Mode="SelectionOnly"
OutFields="*"
Url="http://sampleserver1.arcgisonline.com/ArcGIS/rest/services/Petroleum/KGS_OilGasFields_Kansas/MapServer/0">
<esri:FeatureLayer.Renderer>
<esri:SimpleRenderer>
<esri:SimpleFillSymbol Fill="Blue" />
</esri:SimpleRenderer>
</esri:FeatureLayer.Renderer>
</esri:FeatureLayer>
</esri:Map>
<StackPanel VerticalAlignment="Bottom" HorizontalAlignment="Center">
<Button Content="Select" DataContext="{StaticResource MyEditor}" Command="{Binding Select}"/>
<esri:FeatureDataGrid x:Name="MyFDG" Map="{Binding ElementName=MyMap}" GraphicsLayer="{Binding ElementName=MyMap, Path=Layers[Parcel]}" Height="200"/>
</StackPanel>
</Grid>