The answer from Judah did not work for me (or is not complete) as the application was exiting after the first BeginOutputReadLine();
This works for me as a complete snippet, reading the constant output of a ping:
var process = new Process();
process.StartInfo.FileName = "ping";
process.StartInfo.Arguments = "google.com -t";
process.StartInfo.RedirectStandardOutput = true;
process.StartInfo.UseShellExecute = false;
process.OutputDataReceived += (sender, a) => Console.WriteLine(a.Data);
process.Start();
process.BeginOutputReadLine();
process.WaitForExit();