Why isn't my WPF Datagrid showing data?

Typically in WPF, you would leverage an ObservableCollection<>

Because then you can just .Add() / .Remove() elements to/from the source collection, and any controls bound (Data Binding) automatically get updated (Automatic Property Change Notification). Those are 2 important concepts in WPF.

Main Window View Model

using System.Collections.Generic;

namespace TestDatagrid345.ViewModels
{
  class Window1ViewModel
  {
    private ObservableCollection<Customer> _customers = new ObservableCollection<Customer>();
    public ObservableCollection<Customer> Customers
    {
      Get { return _customers; }
    }
  }
}

Main Window

using System.Collections.Generic;
using System.Windows;

namespace TestDatagrid345
{
  public partial class Window1 : Window
  {
    Window1ViewModel _viewModel;

    public Window1()
    {
      InitializeComponent();
      _viewModel = (Window1ViewModel)this.DataContext; // @#$% (see XAML)
    }

    private void Window_Loaded(object sender, RoutedEventArgs e)
    {
      // but this stuff could instead be done on a 'Submit' button click on form:
      _viewModel.Customers.Add(new Customer { FirstName = "Tom", LastName = "Jones" });
      _viewModel.Customers.Add(new Customer { FirstName = "Joe", LastName = "Thompson" });
      _viewModel.Customers.Add(new Customer { FirstName = "Jill", LastName = "Smith" });
    }
  }
}

Main Window XAML

<Window
  x:Class="TestDatagrid345.MainWindow"
  xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
  xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
  xmlns:vm="clr-namespace:TestDatagrid345.ViewModels"
  Title="Window1"
  Height="350"
  Width="525"
  WindowState="Maximized">
  <Window.DataContext>
    <vm:Window1ViewModel /> <!-- this needs to be here for @#$% -->
  </Window.DataContext>
  <Grid>
   <DataGrid
      AutoGenerateColumns="True"
      ItemsSource="{Binding Path=Customers}"
      AlternatingRowBackground="LightBlue"
      AlternationCount="2" />
  </Grid>
</Window>

You need to add

DataContext = Customers;

In your Window_Loaded()

try it: populate you _customers list and asign property ItemsSource

        dataGrid1.ItemsSource = _customers;