You should be able to do so by using the ItemsSource property of the grid and referencing your collection (probably located in your view model), like this:
ItemsSource="{Binding Path=DownloadsCollection}"
Then add a binding on your columns to show the info (properties) of your MyClass objects in the collection.
For a more detailed tutorial on how to do it, check this link.
Edit:
You can simply try something like this to see if everything works right and then move to custom columns:
<DataGrid ItemsSource="{Binding DownloadsCollection}" />