The manifest is an xml file so you can query it with Linq to XML:
using System.Xml.Linq;
using Windows.ApplicationModel;
using Windows.Storage;
private async void GetInfo(object sender, RoutedEventArgs e)
{
StorageFile file = await Package.Current.InstalledLocation.GetFileAsync("AppxManifest.xml");
string manifestXml = await FileIO.ReadTextAsync(file);
XDocument doc = XDocument.Parse(manifestXml);
XNamespace packageNamespace = "http://schemas.microsoft.com/appx/2010/manifest";
var displayName = (from name in doc.Descendants(packageNamespace + "DisplayName")
select name.Value).First();
var publisherDsplName = (from publisher in doc.Descendants(packageNamespace + "PublisherDisplayName")
select publisher.Value).First();
string output = "DisplayName: " + displayName + ", PublisherDisplayName: " + publisherDsplName;
txtBlock.Text = output;
}