A lot of the answers here were written prior to Roslyn (the open-source .NET C# and VB compilers) moving to .NET 4.6. So they won't help you if your project targets, say, 4.5.2 as mine did (inherited and can't be changed).
But you can grab a previous version of Roslyn from https://www.nuget.org/packages/Microsoft.Net.Compilers and install that instead of the latest version. I used 1.3.2. (I tried 2.0.1 - which appears to be the last version that runs on .NET 4.5 - but I couldn't get it to compile*.) Run this from the Package Manager console in VS 2013:
PM> Install-Package Microsoft.Net.Compilers -Version 1.3.2
Then restart Visual Studio. I had a couple of problems initially; you need to set the C# version back to default (C#6.0 doesn't appear in the version list but seems to have been made the default), then clean, save, restart VS and recompile.
Interestingly, I didn't have any IntelliSense errors due to the C#6.0 features used in the code (which were the reason for wanting C#6.0 in the first place).
* version 2.0.1 threw error The "Microsoft.CodeAnalysis.BuildTasks.Csc task could not be loaded from the assembly Microsoft.Build.Tasks.CodeAnalysis.dll. Could not load file or assembly 'Microsoft.Build.Utilities.Core, Version=14.0.0.0, Culture=neutral, PublicKeyToken=b03f5f7f11d50a3a' or one of its dependencies. The system cannot find the file specified. Confirm that the declaration is correct, that the assembly and all its dependencies are available, and that the task contains a public class that implements Microsoft.Build.Framework.ITask.
UPDATE One thing I've noticed since posting this answer is that if you change any code during debug ("Edit and Continue"), you'll like find that your C#6.0 code will suddenly show as errors in what seems to revert to a pre-C#6.0 environment. This requires a restart of your debug session. VERY annoying especially for web applications.