How to Debug in a standalone build?

There isn't an easy way to get a console for a build, but you can check the log file after you run it. The file will be in the data folder that comes with the build. It should list any errors encountered during runtime.

Something similar. http://unifycommunity.com/wiki/index.php?title=DebugConsole

In other words, you sure can write your own DebugConsole which show up things at your own way. The only downside is you won't be able to double click the line at Unity's Console and get linked back to it in the source code.

Following the question edit, you want then to see the Log Files. It will depend on your platform, please refer to the docs. They're currently at: http://docs.unity3d.com/Documentation/Manual/LogFiles.html

But you still can use try & catch or simple error prevention and something like the DebugConsole. Maybe sending e-mails or a SOAP to your server reporting errors. It's much more recommended.

There´s an easy way too... Just download DbgView (http://technet.microsoft.com/en-us/sysinternals/bb896647.aspx) and use OutputDebugString() to write to it.

C# Example:

 using UnityEngine; 
 using System.Collections;
 using System.Runtime.InteropServices;

 public class Debug : MonoBehaviour 
 {
     [DllImport("kernel32.dll", CharSet = CharSet.Auto)]
     public static extern void OutputDebugString(string message);

     void Start() 
     {
         OutputDebugString("started...");
     }
 }