This can be quite easily achieved using the ProcessStartInfo.RedirectStandardOutput property. A full sample is contained in the linked MSDN documentation; the only caveat is that you may have to redirect the standard error stream as well to see all output of your application.
Process compiler = new Process();
compiler.StartInfo.FileName = "csc.exe";
compiler.StartInfo.Arguments = "/r:System.dll /out:sample.exe stdstr.cs";
compiler.StartInfo.UseShellExecute = false;
compiler.StartInfo.RedirectStandardOutput = true;
compiler.Start();
Console.WriteLine(compiler.StandardOutput.ReadToEnd());
compiler.WaitForExit();