You can also do it like this:

public void RunProcess()
{
    Process process = new Process();
    process.StartInfo.FileName = "consoleApp.exe";

    process.StartInfo.UseShellExecute = false;
    process.StartInfo.RedirectStandardOutput = true;
    process.StartInfo.CreateNoWindow = true;
    process.Start();

    for (; ; )
    {
        string line = process.StandardOutput.ReadLine();
        if (line == null)
            break;

        Dispatcher.Invoke(new Action(() =>
            {
                textBlock1.Text += outLine.Data + Environment.NewLine;
            }), System.Windows.Threading.DispatcherPriority.Normal);
    }
    ...
}