I have a listbox like this:

<ListBox ItemsSource="{Binding Users}" SelectedItem="{Binding CurrentSelectedUser}"
            DisplayMemberPath="Username"/>

Users is a Observable collection of User which is a class with 2 properties which is Username and Password.
I then have a property called CurrentUser on my view model. What I want to do is change the colour of the listboxs item if the Text on it is equal to CurrentUser.Username. Here is what I have tried so far:

<ListBox ItemsSource="{Binding Users}" SelectedItem="{Binding CurrentSelectedUser}"
            DisplayMemberPath="Username">
    <ListBox.ItemContainerStyle>
        <Style BasedOn="{StaticResource {x:Type ListBoxItem}}" TargetType="{x:Type ListBoxItem}">
            <Style.Triggers>
                <DataTrigger Binding="{Binding Content.Username}" Value="{Binding CurrentUser.Username}">
                    <Setter Property="Background" Value="Green"></Setter>
                </DataTrigger>
            </Style.Triggers>
        </Style>
    </ListBox.ItemContainerStyle>
</ListBox>

This doesn't work. Is there any way to do this? I know that Value is not a dependency property. But I want to do something like this.

It's not compiling because value is not a dependency property, said that you cannot use binding in a non dependency property.

You can use IMultiValueConverter to return the color according with the parameter received, here's an example.

Converter:

public class Converter : IMultiValueConverter
{
    public Converter()
    {

    }

    public object Convert(object[] values, Type targetType, object parameter, CultureInfo culture)
    {
        var currentPersonName = values[0].ToString();
        var listItemPersonName = values[1].ToString();

        return currentPersonName == listItemPersonName ? Brushes.Red : Brushes.Black;
    }

    public object[] ConvertBack(object value, Type[] targetTypes, object parameter, CultureInfo culture)
    {
        throw new NotImplementedException();
    }
}

Here you will receive the two names by parameter, so you can compare and return the color you want.

You pass these two values by Multibinding, here's the XAML.

XAML:

 <Window.Resources>
    <local:Converter x:Key="converter"/>
    <Style  x:Key="style" TargetType="ListBoxItem">
        <Setter Property="Foreground">
            <Setter.Value>
                <MultiBinding Converter="{StaticResource converter}">
                    <MultiBinding.Bindings>
                        <Binding Path="DataContext.CurrentPerson.UserName"
                                 RelativeSource="{RelativeSource AncestorType={x:Type Window}}"/>
                        <Binding Path="UserName"/>
                    </MultiBinding.Bindings>
                </MultiBinding>
            </Setter.Value>
        </Setter>
    </Style>
</Window.Resources>
<ListBox ItemsSource="{Binding Persons}"
         DisplayMemberPath="{Binding UserName}"
         ItemContainerStyle="{StaticResource style}"
         SelectedItem="{Binding SelectedPerson}">

</ListBox>

I did a style just like you did, but instead use DataTrigger I used a Multibinding to pass the values to be compared to the converter.

In the first binding I retrieve the userName of the current person in my viewModel, to do this I need specify where is the object, this is the reason of relativeSource.

In the second binding, I just get the Property UserName directly of the ListItemBox DataContext, which has an object of type Person bind to it.

And that is it, it works like expected.