You'll want to use a Style to bind the Background of ListViewItem to the item for the row. The item is the default DataContext of the ListViewItem so this should be straightforward:
<Grid xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:sys="clr-namespace:System;assembly=mscorlib"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml">
<Grid.Resources>
<x:Array x:Key="colors" Type="{x:Type sys:String}">
<sys:String>Red</sys:String>
<sys:String>Yellow</sys:String>
<sys:String>#0000FF</sys:String>
</x:Array>
</Grid.Resources>
<ListView ItemsSource="{StaticResource colors}">
<ListView.Resources>
<Style TargetType="{x:Type ListViewItem}">
<Setter Property="Background" Value="{Binding .}"/>
</Style>
</ListView.Resources>
</ListView>
</Grid>
Instead of binding to the whole item you'll bind to the BackgroundColor, but it should be similar to the above. You have have to use a converter with the binding to prefix a "#", this is the signal to the built-in BrushConverter to parse it as hex.