Your post is really good, you just skip the double quotes on the Source property
ListView BackgroundColor " Transparent:
<RelativeLayout>
<Image Source="background.png"
BackgroundColor="Transparent"
VerticalOptions="CenterAndExpand"
HorizontalOptions="CenterAndExpand"/>
<ListView x:Name="listView"
VerticalOptions="CenterAndExpand"
HorizontalOptions="CenterAndExpand"
BackgroundColor="Transparent"
ItemTapped="OnItemTapped"
ItemsSource="{Binding .}" />
</RelativeLayout>