To add to @Craig Schwarze answer,
Here are some related MSDN links:
Loading and Running a Local Package Programmatically:
Loading and Running a Remote Package Programmatically
Capturing Events from a Running Package:
using System;
using Microsoft.SqlServer.Dts.Runtime;
namespace RunFromClientAppWithEventsCS
{
  class MyEventListener : DefaultEvents
  {
    public override bool OnError(DtsObject source, int errorCode, string subComponent, 
      string description, string helpFile, int helpContext, string idofInterfaceWithError)
    {
      // Add application-specific diagnostics here.
      Console.WriteLine("Error in {0}/{1} : {2}", source, subComponent, description);
      return false;
    }
  }
  class Program
  {
    static void Main(string[] args)
    {
      string pkgLocation;
      Package pkg;
      Application app;
      DTSExecResult pkgResults;
      MyEventListener eventListener = new MyEventListener();
      pkgLocation =
        @"C:\Program Files\Microsoft SQL Server\100\Samples\Integration Services" +
        @"\Package Samples\CalculatedColumns Sample\CalculatedColumns\CalculatedColumns.dtsx";
      app = new Application();
      pkg = app.LoadPackage(pkgLocation, eventListener);
      pkgResults = pkg.Execute(null, null, eventListener, null, null);
      Console.WriteLine(pkgResults.ToString());
      Console.ReadKey();
    }
  }
}